4834:
3491:
5055:
4614:
8128:. All of these projections (including the Perron projection) have the same positive diagonal, moreover choosing any one of them and then taking the modulus of every entry invariably yields the Perron projection. Some donkey work is still needed in order to establish the cyclic properties (6)–(8) but it's essentially just a matter of turning the handle. The spectral decomposition of
1119:
4213:
3252:
1582:
8628:
4877:
4829:{\displaystyle M=\left({\begin{smallmatrix}0&1&0&0&\cdots &0\\0&0&1&0&\cdots &0\\0&0&0&1&\cdots &0\\\vdots &\vdots &\vdots &\vdots &&\vdots \\0&0&0&0&\cdots &1\\1&1&0&0&\cdots &0\end{smallmatrix}}\right)}
6192:. This gives another proof that there are no eigenvalues which have greater absolute value than Perron–Frobenius one. It also contradicts the existence of the Jordan cell for any eigenvalue which has absolute value equal to 1 (in particular for the Perron–Frobenius one), because existence of the Jordan cell implies that
816:
6345:
6432:
Other eigenvectors must contain negative or complex components since eigenvectors for different eigenvalues are orthogonal in some sense, but two positive eigenvectors cannot be orthogonal, so they must correspond to the same eigenvalue, but the eigenspace for the Perron–Frobenius is one-dimensional.
1273:
2560:
All statements of the Perron–Frobenius theorem for positive matrices remain true for primitive matrices. The same statements also hold for a non-negative irreducible matrix, except that it may possess several eigenvalues whose absolute value is equal to its spectral radius, so the statements need to
6179:
3486:{\displaystyle PAP^{-1}={\begin{pmatrix}O&A_{1}&O&O&\ldots &O\\O&O&A_{2}&O&\ldots &O\\\vdots &\vdots &\vdots &\vdots &&\vdots \\O&O&O&O&\ldots &A_{h-1}\\A_{h}&O&O&O&\ldots &O\end{pmatrix}},}
4035:
7946:. The spectral projections aren't neatly blocked as in the Jordan form. Here they are overlaid and each generally has complex entries extending to all four corners of the square matrix. Nevertheless, they retain their mutual orthogonality which is what facilitates the decomposition.
1755:
There is an extension to matrices with non-negative entries. Since any non-negative matrix can be obtained as a limit of positive matrices, one obtains the existence of an eigenvector with non-negative components; the corresponding eigenvalue will be non-negative and greater than
8467:
8372:
5050:{\displaystyle \left({\begin{smallmatrix}B_{1}&*&*&\cdots &*\\0&B_{2}&*&\cdots &*\\\vdots &\vdots &\vdots &&\vdots \\0&0&0&\cdots &*\\0&0&0&\cdots &B_{h}\end{smallmatrix}}\right)}
3916:
8458:
8280:
7593:
2001:
has strictly positive components (in contrast with the general case of non-negative matrices, where components are only non-negative). Also all such eigenvalues are simple roots of the characteristic polynomial. Further properties are described below.
9959:
6205:
5424:). Hence, there exists an eigenvalue λ on the unit circle, and all the other eigenvalues are less or equal 1 in absolute value. Suppose that another eigenvalue λ ≠ 1 also falls on the unit circle. Then there exists a positive integer
5287:(where it is the matrix-theoretic equivalent of the convergence of an irreducible finite Markov chain to its stationary distribution, formulated in terms of the transition matrix of the chain; see, for example, the article on the
1114:{\displaystyle r=\sup _{x>0}\inf _{y>0}{\frac {y^{\top }Ax}{y^{\top }x}}=\inf _{x>0}\sup _{y>0}{\frac {y^{\top }Ax}{y^{\top }x}}=\inf _{x>0}\sup _{y>0}\sum _{i,j=1}^{n}y_{i}a_{ij}x_{j}/\sum _{i=1}^{n}y_{i}x_{i}.}
4436:
1884:
1819:
4844:
Numerous books have been written on the subject of non-negative matrices, and Perron–Frobenius theory is invariably a central feature. The following examples given below only scratch the surface of its vast application domain.
5677:
The proof requires two additional arguments. First, the power method converges for matrices which do not have several eigenvalues of the same absolute value as the maximal one. The previous section's argument guarantees this.
5986:
1897:
matrices — for which a non-trivial generalization is possible. For such a matrix, although the eigenvalues attaining the maximal absolute value might not be unique, their structure is under control: they have the form
8642:
is an example of a primitive matrix with zero diagonal. If the diagonal of an irreducible non-negative square matrix is non-zero then the matrix must be primitive but this example demonstrates that the converse is false.
3823:
2193:
1679:
7480:
7263:
4208:{\displaystyle PA^{q}P^{-1}={\begin{pmatrix}A_{1}&O&O&\dots &O\\O&A_{2}&O&\dots &O\\\vdots &\vdots &\vdots &&\vdots \\O&O&O&\dots &A_{d}\\\end{pmatrix}}}
1577:{\displaystyle r=\sup _{z>0}\ \inf _{x>0,\ y>0,\ x\circ y=z}{\frac {y^{\top }Ax}{y^{\top }x}}=\sup _{z>0}\ \inf _{x>0,\ y>0,\ x\circ y=z}\sum _{i,j=1}^{n}y_{i}a_{ij}x_{j}/\sum _{i=1}^{n}y_{i}x_{i}.}
1260:
8126:
661:
7644:
8623:{\displaystyle \left({\begin{smallmatrix}0&1&0&0&0\\1&0&0&0&0\\0&0&0&1&0\\0&0&0&0&1\\0&0&1&0&0\end{smallmatrix}}\right)}
5221:
is a square matrix each of whose rows (columns) consists of non-negative real numbers whose sum is unity. The theorem cannot be applied directly to such matrices because they need not be irreducible.
9079:
Donsker, M.D. and
Varadhan, S.S., 1975. On a variational formula for the principal eigenvalue for operators with maximum principle. Proceedings of the National Academy of Sciences, 72(3), pp.780-783.
7801:. The trick here is to split the Perron root from the other eigenvalues. The spectral projection associated with the Perron root is called the Perron projection and it enjoys the following property:
7715:
7154:
7318:
5681:
Second, to ensure strict positivity of all of the components of the eigenvector for the case of irreducible matrices. This follows from the following fact, which is of independent interest:
8289:
5861:
This section proves that the Perron–Frobenius eigenvalue is a simple root of the characteristic polynomial of the matrix. Hence the eigenspace associated to Perron–Frobenius eigenvalue
5529:
Absolutely the same arguments can be applied to the case of primitive matrices; we just need to mention the following simple lemma, which clarifies the properties of primitive matrices.
3833:
8381:
8203:
7340:
7183:
2713:
7489:
2452:
2411:
2382:
2348:
2319:
8967:
3584:
3065:
336:
9070:
Birkhoff, Garrett and Varga, Richard S., 1958. Reactor criticality and nonnegative matrices. Journal of the
Society for Industrial and Applied Mathematics, 6(4), pp.354-377.
7362:
3122:
3551:
3172:
2850:
2828:
6340:{\displaystyle J^{k}={\begin{pmatrix}\lambda &1\\0&\lambda \end{pmatrix}}^{k}={\begin{pmatrix}\lambda ^{k}&k\lambda ^{k-1}\\0&\lambda ^{k}\end{pmatrix}},}
3693:
298:
3639:
2674:
236:
2616:
262:
1919:
1745:
428:
7807:
Perron's findings and also (1)–(5) of the theorem are corollaries of this result. The key point is that a positive projection always has rank one. This means that if
3192:
1961:
3221:
8638:, thus when the original matrix is reducible the projections may lose non-negativity and there is no chance of expressing them as limits of its powers. The matrix
1589:
3713:
3660:
3518:
3241:
3142:
3085:
3021:
3001:
2981:
2957:
2937:
2917:
2894:
2870:
2806:
2784:
2760:
2733:
2636:
2590:
1998:
1940:
1148:
6790:
9000:
6409:
is simple root of the characteristic polynomial. In the case of nonprimitive matrices, there exist other eigenvalues which have the same absolute value as
6174:{\displaystyle \|v\|_{\infty }=\|A^{k}v\|_{\infty }\geq \|A^{k}\|_{\infty }\min _{i}(v_{i}),~~\Rightarrow ~~\|A^{k}\|_{\infty }\leq \|v\|/\min _{i}(v_{i})}
4313:
1828:
1763:
95:); and even to ranking of American football teams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is
7049:. Trace of projector equals the dimension of its image. It was proved before that it is not more than one-dimensional. From the definition one sees that
7990:). It may then be shown that the peripheral projection of an irreducible non-negative square matrix is a non-negative matrix with a positive diagonal.
8706:
The nonnegative eigenvector is often normalized so that the sum of its components is equal to unity; in this case, the eigenvector is the vector of a
8630:
provide simple examples of what can go wrong if the necessary conditions are not met. It is easily seen that the Perron and peripheral projections of
5267:≠ 0. If the underlying graph of such a matrix is strongly connected, then the matrix is irreducible, and thus the theorem applies. In particular, the
9738:
3733:
2120:
7958:
is irreducible and non-negative is broadly similar. The Perron projection is still positive but there may now be other eigenvalues of modulus ρ(
7367:
7188:
10012:(1959 edition had different title: "Applications of the theory of matrices". Also the numeration of chapters is different in the two editions.)
8791: – self-adjoint (or Hermitian) element A of a C*-algebra A is called positive if its spectrum σ (A) consists of non-negative real numbers
9751:
2549:(see Kitchens page 16). The period is also called the index of imprimitivity (Meyer page 674) or the order of cyclicity. If the period is 1,
10299:. 2nd rev. ed., 1981, XVI, 288 p., Softcover Springer Series in Statistics. (Originally published by Allen & Unwin Ltd., London, 1973)
5232:) by the remark above. It might not be the only eigenvalue on the unit circle: and the associated eigenspace can be multi-dimensional. If
1890:= 0, which is not a simple root of the characteristic polynomial, and the corresponding eigenvector (1, 0) is not strictly positive.
5329:, and the lesser eigenvalues to the decay modes of a system that is not in equilibrium. Thus, the theory offers a way of discovering the
8061:
9873:
9821:
9620:
9568:
9501:
9449:
9385:
9229:
9176:
8942:
8890:
6402:), so it also tends to infinity. The resulting contradiction implies that there are no Jordan cells for the corresponding eigenvalues.
7811:
is an irreducible non-negative square matrix then the algebraic and geometric multiplicities of its Perron root are both one. Also if
10279:
592:
2763:
9259:
7598:
5209:). While there will still be eigenvectors with non-negative components it is quite possible that none of these will be positive.
5228:
is row-stochastic then the column vector with each entry 1 is an eigenvector corresponding to the eigenvalue 1, which is also ρ(
676:
168:). The Perron–Frobenius theorem describes the properties of the leading eigenvalue and of the corresponding eigenvectors when
10304:
10092:
10033:
10006:
9719:
9694:
9666:
9323:
5373:(Perron–Frobenius eigenvalue or Perron root), which is strictly greater in absolute value than all other eigenvalues, hence
5333:
in what would otherwise appear to be reversible, deterministic dynamical processes, when examined from the point of view of
5670:. Hence the limiting vector is also non-negative. By the power method this limiting vector is the dominant eigenvector for
10373:
8663:
A problem that causes confusion is a lack of standardisation in the definitions. For example, some authors use the terms
7659:
7098:
4853:
The Perron–Frobenius theorem does not apply directly to non-negative matrices. Nevertheless, any reducible square matrix
10079:
9842:
9790:
9589:
9537:
9470:
9418:
9354:
9198:
9145:
8911:
8859:
3956:
2540:
is irreducible, the period can be defined as the greatest common divisor of the lengths of the closed directed paths in
1689:
9996:
9518:
9246:
8977:
7279:
6797:, and is called the Perron projection. The above assertion is not true for general non-negative irreducible matrices.
5309:, which, in many ways, resemble finite-dimensional matrices. These are commonly studied in physics, under the name of
10289:
10237:
10118:
9296:
9269:
8367:{\displaystyle \left({\begin{smallmatrix}1&0&0\\1&0&0\\\!\!\!-1&1&1\end{smallmatrix}}\right)}
6808:
which is strictly greater than the other eigenvalues in absolute value and is the simple root of the characteristic
5932:
non-negativity implies strict positivity for any eigenvector. On the other hand, as above at least one component of
1703:
are sometimes normalized so that the sum of their components is equal to 1; in this case, they are sometimes called
5236:
is row-stochastic and irreducible then the Perron projection is also row-stochastic and all its rows are equal.
3911:{\displaystyle A=\left({\begin{smallmatrix}0&0&1\\0&0&1\\1&1&0\end{smallmatrix}}\right)}
55:
can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of
7798:
6425:, the Perron–Frobenius eigenvector is the only (up to multiplication by constant) non-negative eigenvector for
10311:
8453:{\displaystyle \left({\begin{smallmatrix}0&1&1\\1&0&1\\1&1&0\end{smallmatrix}}\right)}
8275:{\displaystyle \left({\begin{smallmatrix}1&0&0\\1&0&0\\1&1&1\end{smallmatrix}}\right)}
749:
A "Min-max" Collatz–Wielandt formula takes a form similar to the one above: for all strictly positive vectors
10321:
8655:
is block-diagonal cyclic, then the eigenvalues are {1,-1} for the first block, and {1,ω,ω} for the lower one
7588:{\displaystyle \scriptstyle \left\|A\right\|_{\infty }=\max \limits _{1\leq i\leq m}\sum _{j=1}^{n}|a_{ij}|.}
8817:
Bowles, Samuel (1981-06-01). "Technical change and the profit rate: a simple proof of the
Okishio theorem".
9100:"A Trace Inequality for M-matrices and the Symmetrizability of a Real Matrix by a Positive Diagonal Matrix"
7720:
This fact is specific to non-negative matrices; for general matrices there is nothing similar. Given that
7323:
7166:
5272:
2683:
2267:
8647:
is an example of a matrix with several missing spectral teeth. If ω = e then ω = 1 and the eigenvalues of
10316:
6707:
5346:
80:
8651:
are {1,ω,ω=-1,ω} with a dimension 2 eigenspace for +1 so ω and ω are both absent. More precisely, since
5353:–Wielandt formula described above to extend and clarify Frobenius's work. Another proof is based on the
2557:. It can be proved that primitive matrices are the same as irreducible aperiodic non-negative matrices.
9911:
2455:
2416:
5697:, then it is necessarily strictly positive and the corresponding eigenvalue is also strictly positive.
2387:
2353:
2324:
2290:
10378:
7266:
5322:
3558:
3033:
2458:
if there were no non-trivial invariant subspaces at all, not only considering coordinate subspaces.)
446:
303:
8752: – Variational characterization of eigenvalues of compact Hermitian operators on Hilbert spaces
7345:
8797:
8707:
5526:) = 1. This contradiction means that λ=1 and there can be no other eigenvalues on the unit circle.
5360:
5300:
3094:
1128:
72:
9097:
5701:
Proof. One of the definitions of irreducibility for non-negative matrices is that for all indexes
3534:
3147:
2833:
2811:
125:
real numbers as elements and matrices with exclusively non-negative real numbers as elements. The
9088:
Friedland, S., 1981. Convex spectral functions. Linear and multilinear algebra, 9(4), pp.299-316.
5656:
can be chosen arbitrarily except for some measure zero set). Starting with a non-negative vector
5369:
is a positive (or more generally primitive) matrix, then there exists a real positive eigenvalue
5288:
3665:
2507:
267:
7930:
The power method is a convenient way to compute the Perron projection of a primitive matrix. If
3591:
2644:
199:
10368:
8755:
5245:
5160:) is equal to the identity plus a nilpotent matrix. But such a matrix is always invertible (if
3918:
shows that the (square) zero-matrices along the diagonal may be of different sizes, the blocks
2595:
241:
8959:
1901:
1714:
580:
There are no other positive (moreover non-negative) eigenvectors except positive multiples of
404:
9916:
7938:
are the positive row and column vectors that it generates then the Perron projection is just
3177:
1946:
9655:
3200:
2561:
be correspondingly modified. In fact the number of such eigenvalues is equal to the period.
8782:
5727:-th is strictly positive, the corresponding eigenvalue is strictly positive, indeed, given
2284:
141:
118:
76:
56:
6897:
The projection and commutativity properties are elementary corollaries of the definition:
4222:
are irreducible matrices having the same maximal eigenvalue. The number of these matrices
8:
7804:
The Perron projection of an irreducible non-negative square matrix is a positive matrix.
2023:
10050:
Keener, James (1993), "The Perron–Frobenius theorem and the ranking of football teams",
4304:
2876:
eigenvectors whose components are all positive are those associated with the eigenvalue
461:
is one-dimensional. (The same is true for the left eigenspace, i.e., the eigenspace for
10225:
10207:
10169:
10067:
10039:
9943:
9891:
9682:
9646:
8994:
8830:
8703:. Then irreducible non-negative square matrices and connected matrices are synonymous.
5388:
This statement does not hold for general non-negative irreducible matrices, which have
5334:
4307:
3698:
3645:
3503:
3226:
3127:
3070:
3006:
2986:
2983:. Each of them is a simple root of the characteristic polynomial and is the product of
2966:
2942:
2922:
2902:
2879:
2855:
2791:
2769:
2745:
2718:
2621:
2575:
2108:
1983:
1925:
145:
108:
44:
7483:
7273:
5951:
586:), i.e., all other eigenvectors must have at least one negative or non-real component.
10300:
10285:
10233:
10211:
10173:
10114:
10088:
10029:
10002:
9947:
9887:
9867:
9815:
9715:
9690:
9662:
9614:
9562:
9495:
9443:
9379:
9329:
9319:
9292:
9265:
9223:
9170:
9116:
9099:
8973:
8936:
8884:
8834:
8788:
5310:
5218:
4431:{\displaystyle \lim _{k\rightarrow \infty }1/k\sum _{i=0,...,k}A^{i}/r^{i}=(vw^{T}),}
1976:
1879:{\displaystyle A=\left({\begin{smallmatrix}0&1\\0&0\end{smallmatrix}}\right)}
1814:{\displaystyle A=\left({\begin{smallmatrix}0&1\\1&0\end{smallmatrix}}\right)}
1124:
68:
8691:
is an overloaded term. For avoidance of doubt a non-zero non-negative square matrix
5900:
has a basis consisting of real vectors.) Assuming at least one of the components of
2454:
has no non-trivial invariant coordinate subspaces. (By comparison, this would be an
2277:
is the field of real or complex numbers, then we also have the following condition.
10346:
10197:
10189:
10161:
10136:
10059:
10043:
10021:
9992:
9933:
9925:
9766:
9111:
8826:
7784: = (1, 1, ..., 1) and immediately obtains the inequality.
5582:
Applying the same arguments as above for primitive matrices, prove the main claim.
5326:
5306:
5268:
4454:
1265:
1132:
800:
796:
88:
8724:
are alternative names for the Perron root. Spectral projections are also known as
3520:
denotes a zero matrix and the blocks along the main diagonal are square matrices.
434:. If the matrix coefficients are algebraic, this implies that the eigenvalue is a
9980:
9968:
9954:
9098:
Miroslav
Fiedler; Charles R. Johnson; Thomas L. Markham; Michael Neumann (1985).
8749:
7773:
and this observation can be extended to all non-negative matrices by continuity.
5943:
Case: There are no Jordan cells corresponding to the Perron–Frobenius eigenvalue
5500:
5378:
5354:
2639:
2567:
2051:
399:
181:
137:
122:
36:
8477:
8391:
8299:
8213:
5361:
Perron root is strictly maximal eigenvalue for positive (and primitive) matrices
4887:
4630:
4028:
is reducible, then it is completely reducible, i.e. for some permutation matrix
3849:
1844:
1779:
10149:
9770:
8778:
8772:
7962:) that negate use of the power method and prevent the powers of (1 −
7777:
5350:
3525:
3088:
2217:
691:
383:
379:
165:
161:
10202:
10025:
9938:
9049:
7724:
is positive (not just non-negative), then there exists a positive eigenvector
4579:
is positive. Moreover, this is the best possible result, since for the matrix
10362:
10152:(1942), "Einschließungssatz für die charakteristischen Zahlen von Matrizen",
10098:
9849:
9838:
9797:
9786:
9596:
9585:
9544:
9533:
9477:
9466:
9425:
9414:
9361:
9350:
9333:
9205:
9193:
9152:
9141:
8963:
8918:
8907:
8866:
8855:
8838:
5330:
3024:
2062:. More explicitly, for any linear subspace spanned by standard basis vectors
2055:
1968:
435:
130:
96:
84:
20:
9029:
Landau, Edmund (1895), "Zur relativen
Wertbemessung der Turnierresultaten",
8775: – Algebraic matrix element to analyze a polynomial by its coefficients
6405:
Combining the two claims above reveals that the Perron–Frobenius eigenvalue
5685:
Lemma: given a positive (or more generally irreducible non-negative) matrix
3818:{\displaystyle \min _{i}\sum _{j}a_{ij}\leq r\leq \max _{i}\sum _{j}a_{ij}.}
2564:
Results for non-negative matrices were first obtained by
Frobenius in 1912.
2188:{\displaystyle PAP^{-1}\neq {\begin{pmatrix}E&F\\O&G\end{pmatrix}},}
1893:
However, Frobenius found a special subclass of non-negative matrices —
1674:{\displaystyle \min _{i}\sum _{j}a_{ij}\leq r\leq \max _{i}\sum _{j}a_{ij}.}
10247:
10243:
9907:
6621:. For the opposite inequality, we consider an arbitrary nonnegative vector
5928:
is an eigenvector. It is non-negative, hence by the lemma described in the
5609:, which states that for a sufficiently generic (in the sense below) matrix
5606:
5318:
5284:
173:
64:
28:
9261:
Symbolic dynamics: one-sided, two-sided and countable state markov shifts.
7475:{\displaystyle \scriptstyle \|A\|\geq |Ax|/|x|=|\lambda x|/|x|=|\lambda |}
7258:{\displaystyle \scriptstyle |\lambda |\;\leq \;\max _{i}\sum _{j}|a_{ij}|}
9650:
9284:
8800: – A generalization of the Perron–Frobenius theorem to Banach spaces
8769: – Complex square matrix for which every principal minor is positive
5642:
5595:
52:
4555:. To check primitivity, one needs a bound on how large the minimal such
2528:
is irreducible, the period of every index is the same and is called the
1760:, in absolute value, to all other eigenvalues. However, for the example
1255:{\displaystyle r=\sup _{p}\inf _{x>0}\sum _{i=1}^{n}p_{i}_{i}/x_{i}.}
10193:
10165:
10141:
10071:
9985:
Sitzungsberichte der Königlich
Preussischen Akademie der Wissenschaften
9973:
Sitzungsberichte der Königlich
Preussischen Akademie der Wissenschaften
9960:
Sitzungsberichte der Königlich
Preussischen Akademie der Wissenschaften
9929:
9017:
6809:
5787:
is strictly positive. The eigenvector is strict positivity. Then given
5674:, proving the assertion. The corresponding eigenvalue is non-negative.
5646:
454:
126:
60:
48:
1825:= 1 has the same absolute value as the other eigenvalue −1; while for
9313:
6536:
Given a positive (or more generally irreducible non-negative matrix)
188:) found their extension to certain classes of non-negative matrices.
51:
of largest magnitude and that eigenvalue is real. The corresponding
10063:
6800:
Actually the claims above (except claim 5) are valid for any matrix
5865:
is one-dimensional. The arguments here are close to those in Meyer.
1684:
All of these properties extend beyond strictly positive matrices to
553:> 0.) It is known in the literature under many variations as the
10275:, Cambridge Tracts in Mathematics 189, Cambridge Univ. Press, 2012.
8969:
Google's PageRank and Beyond: The
Science of Search Engine Rankings
8766:
8761:
7871:). Thus the only positive eigenvectors are those associated with ρ(
6737:
is one-dimensional and spanned by the Perron–Frobenius eigenvector
6421:
Given positive (or more generally irreducible non-negative matrix)
5590:
For a positive (or more generally irreducible non-negative) matrix
92:
10180:
Wielandt, Helmut (1950), "Unzerlegbare, nicht negative Matrizen",
10127:
Romanovsky, V. (1933), "Sur les zéros des matrices stocastiques",
8121:{\displaystyle \scriptstyle h^{-1}\sum _{1}^{h}\lambda ^{-k}R^{k}}
5962:, but that contradicts the existence of the positive eigenvector.
5853:
is strictly positive, i.e., the eigenvector is strictly positive.
2206:
are non-trivial (i.e. of size greater than zero) square matrices.
6583:
5349:. Another popular method is that of Wielandt (1950). He used the
5283:
The theorem has a natural interpretation in the theory of finite
4468:
be the Perron–Frobenius eigenvalue, then the adjoint matrix for (
3716:
1692:
claims 8.2.11–15 page 667 and exercises 8.2.5,7,9 pages 668–669.
785:
739:
59:. This theorem has important applications to probability theory (
9311:
8758: – Square matrix whose off-diagonal entries are nonpositive
4282:
in which only the non-zero terms are listed, then the period of
2715:
is a positive real number and it is an eigenvalue of the matrix
172:
is a non-negative real square matrix. Early results were due to
8058:
at the eigenvalue λ on the unit circle is given by the formula
7079:
5305:
More generally, it can be extended to the case of non-negative
9687:
A Combinatorial Approach to Matrix Theory and Its Applications
7159:
This is not specific to non-negative matrices: for any matrix
5947:
and all other eigenvalues which have the same absolute value.
5077:
is a square matrix that is either irreducible or zero. Now if
2872:
and whose components are all positive. Moreover these are the
2568:
Perron–Frobenius theorem for irreducible non-negative matrices
656:{\displaystyle \lim _{k\rightarrow \infty }A^{k}/r^{k}=vw^{T}}
6812:. (These requirements hold for primitive matrices as above).
6680:
be a positive (or more generally, primitive) matrix, and let
10259:
Nonnegative Matrices and Applicable Topics in Linear Algebra
9957:(May 1912), "Ueber Matrizen aus nicht negativen Elementen",
6637:(componentwise). Now, we use the positive right eigenvector
6413:. The same claim is true for them, but requires more work.
5929:
4857:
may be written in upper-triangular block form (known as the
7915:
increases the second of these terms decays to zero leaving
7839:
and every row is a positive left eigenvector. Moreover, if
7639:{\displaystyle \scriptstyle \|A\|_{\infty }\geq |\lambda |}
6520:. Since the eigenspace for the Perron–Frobenius eigenvalue
5321:). In this case, the leading eigenvalue corresponds to the
5252:-square matrix is the graph with vertices numbered 1, ...,
2107:
cannot be conjugated into block upper triangular form by a
1707:. Often they are normalized so that the right eigenvector
1586:
The Perron–Frobenius eigenvalue satisfies the inequalities
9586:
chapter 8 example 8.3.4 page 679 and exercise 8.3.9 p. 685
5585:
5435:
is negative. Let ε be half the smallest diagonal entry of
5195:
may be deduced by applying the theorem to the irreducible
3724:
The Perron–Frobenius eigenvalue satisfies the inequalities
2050:
subspaces. Here a non-trivial coordinate subspace means a
532:. (Respectively, there exists a positive left eigenvector
10342:
at the end of the statement of the theorem is incorrect.)
6886:, so the limit exists. The same method works for general
5977:
be a Perron–Frobenius strictly positive eigenvector, so
5416:
be a positive matrix, assume that its spectral radius ρ(
8962:
7982:
corresponding to all the eigenvalues that have modulus
5719:
is strictly positive. Given a non-negative eigenvector
5202:. For example, the Perron root is the maximum of the ρ(
2808:
has both a right and a left eigenvectors, respectively
9312:
Gradshtein, Izrailʹ Solomonovich (18 September 2014).
8065:
7602:
7493:
7371:
7349:
7327:
7283:
7192:
7170:
6664:
6273:
6228:
4073:
3283:
2151:
8671:
to mean > 0 and ≥ 0 respectively. In this article
8470:
8384:
8292:
8206:
8064:
7662:
7601:
7492:
7370:
7348:
7326:
7282:
7191:
7169:
7101:
6416:
6208:
5989:
5107:(if it exists) must have diagonal blocks of the form
4880:
4617:
4316:
4038:
3836:
3736:
3701:
3668:
3648:
3594:
3561:
3537:
3506:
3255:
3229:
3203:
3180:
3174:(i.e. to rotations of the complex plane by the angle
3150:
3130:
3097:
3073:
3036:
3009:
2989:
2969:
2945:
2925:
2905:
2882:
2858:
2836:
2814:
2794:
2772:
2748:
2721:
2686:
2647:
2624:
2598:
2578:
2419:
2390:
2356:
2327:
2293:
2123:
2037:
if any of the following equivalent properties holds.
1986:
1949:
1928:
1904:
1831:
1766:
1717:
1592:
1276:
1151:
819:
595:
407:
306:
270:
244:
202:
9983:(1909), "Über Matrizen aus positiven Elementen, 2",
9971:(1908), "Über Matrizen aus positiven Elementen, 1",
9712:
Time's Arrow: The origins of thermodynamic behaviour
8793:
Pages displaying wikidata descriptions as a fallback
9759:
Mathematical Proceedings of the Royal Irish Academy
7769:. Thus the minimum row sum gives a lower bound for
7710:{\displaystyle \min _{i}\sum _{j}a_{ij}\;\leq \;r.}
7149:{\displaystyle r\;\leq \;\max _{i}\sum _{j}a_{ij}.}
6823:is conjugate to a diagonal matrix with eigenvalues
5575:is entirely zero, but in this case the same row of
5483:this point lies outside the unit disk consequently
5451:which is yet another positive matrix. Moreover, if
2257:is irreducible if and only if its associated graph
9654:
8966:; Langville, Amy N.; Meyer, Carl D. (2006-07-23).
8622:
8452:
8366:
8274:
8120:
7709:
7638:
7587:
7474:
7356:
7334:
7312:
7257:
7177:
7148:
6339:
6173:
5571:, so it can have zero element only if some row of
5049:
4828:
4430:
4207:
3910:
3817:
3707:
3687:
3654:
3633:
3578:
3545:
3512:
3485:
3235:
3215:
3186:
3166:
3136:
3116:
3079:
3059:
3015:
2995:
2975:
2951:
2931:
2911:
2888:
2864:
2844:
2822:
2800:
2778:
2766:. Both right and left eigenspaces associated with
2754:
2727:
2707:
2668:
2630:
2610:
2584:
2446:
2405:
2376:
2342:
2313:
2187:
1992:
1955:
1934:
1913:
1878:
1813:
1739:
1673:
1576:
1254:
1113:
655:
422:
330:
292:
256:
230:
10230:Nonnegative Matrices in the Mathematical Sciences
8338:
8337:
8336:
7313:{\displaystyle \scriptstyle \|A\|\geq |\lambda |}
5392:eigenvalues with the same absolute eigenvalue as
4483:has at least one non-zero diagonal element, then
10360:
9752:"A Spectral Theoretic Proof of Perron–Frobenius"
9681:
8999:: CS1 maint: bot: original URL status unknown (
8949:
7664:
7212:
7111:
6544:on the set of all non-negative non-zero vectors
6146:
6062:
5598:is real and strictly positive (for non-negative
5495:are positive and less than or equal to those in
5491:) > 1. On the other hand, all the entries in
4318:
3780:
3738:
1979:of the matrix. The eigenvector corresponding to
1636:
1594:
1414:
1395:
1303:
1284:
1169:
1159:
987:
971:
915:
899:
843:
827:
597:
10292:(2nd edition, Cambridge University Press, 2009)
9291:. New York: John Wiley & Sons. p. 6 .
6994:, which is one-dimensional by the assumptions.
6464:– is the left Perron–Frobenius eigenvector for
5916:is non-negative, then one of the components of
5357:from which part of the arguments are borrowed.
4520:is irreducible, then the inequality is strict:
10015:
9886:For surveys of results on irreducibility, see
8955:
5723:, and that at least one of its components say
5663:produces the sequence of non-negative vectors
5407:
5191:Therefore, many of the spectral properties of
5133:be the block-diagonal matrix corresponding to
1688:(see below). Facts 1–7 can be found in Meyer
10129:Bulletin de la Société Mathématique de France
8982:. Archived from the original on July 10, 2014
8732:. The period is sometimes referred to as the
8014:is the peripheral projection then the matrix
7970:decaying as in the primitive case whenever ρ(
6605:. Inserting the Perron-Frobenius eigenvector
6524:is one-dimensional, non-negative eigenvector
5081:is non-negative then so too is each block of
4539:is primitive provided it is non-negative and
3947:be an irreducible non-negative matrix, then:
2480:th power is positive for some natural number
2005:
9645:
7610:
7603:
7378:
7372:
7290:
7284:
7080:Inequalities for Perron–Frobenius eigenvalue
6137:
6131:
6119:
6105:
6052:
6038:
6026:
6009:
5997:
5990:
1943:is a real strictly positive eigenvalue, and
663:, where the left and right eigenvectors for
584:(respectively, left eigenvectors except ww'w
6531:
6528:is a multiple of the Perron–Frobenius one.
5432:is a positive matrix and the real part of λ
4571:is a non-negative primitive matrix of size
441:The Perron–Frobenius eigenvalue is simple:
10309:
10126:
10081:Matrix analysis and applied linear algebra
9991:
9734:
9514:
9398:
9242:
7700:
7696:
7210:
7206:
7109:
7105:
5248:. The "underlying graph" of a nonnegative
4848:
4438:where the left and right eigenvectors for
2963:) complex eigenvalues with absolute value
180:) and concerned positive matrices. Later,
10201:
10140:
9979:
9967:
9953:
9937:
9115:
9050:"Über Preisverteilung bei Spielturnieren"
7595:Hence the desired inequality is exactly
7061:. So it is one-dimensional. So choosing (
5239:
2695:
2393:
2361:
2330:
2298:
185:
40:
10179:
9340:
9315:Table of integrals, series, and products
9257:
9251:
9133:
9131:
9129:
9127:
7949:
7780:-Wielandt formula. One takes the vector
7486:of a matrix is the maximum of row sums:
7269:. However another proof is more direct:
7265:. This is an immediate corollary of the
7029:is spanned by it. Respectively, rows of
5936:is zero. The contradiction implies that
5089:is just the union of the spectra of the
3530:: for all non-negative non-zero vectors
1750:
696:: for all non-negative non-zero vectors
85:Leslie population age distribution model
10297:Non-negative matrices and Markov chains
10148:
9007:
8897:
8845:
8679:means ≥ 0. Another vexed area concerns
6651:ξ w x = w ξx ≤ w (Ax) = (w A)x = r w x
6593:For the proof we denote the maximum of
6199:is unbounded. For a two by two matrix:
5880:with the same eigenvalue. (The vectors
5586:Power method and the positive eigenpair
5278:
3223:then there exists a permutation matrix
2676:. Then the following statements hold.
2495:be real and non-negative. Fix an index
2098:is not contained in the same subspace.
742:over all non-negative non-zero vectors
156:→ ∞ is controlled by the eigenvalue of
10361:
10281:Markov Chains and Stochastic Stability
10261:, John Wiley&Sons, New York, 1987.
10049:
9906:
9709:
9047:
9028:
9013:
8816:
7978:, which is the spectral projection of
6882:)), which tends to (1,0,0,...,0), for
5868:Given a strictly positive eigenvector
5345:A common thread in many proofs is the
5212:
4286:equals the greatest common divisor of
3963:, this is also a sufficient condition.
338:. Then the following statements hold.
177:
32:
10345:
10077:
9872:: CS1 maint: archived copy as title (
9834:
9820:: CS1 maint: archived copy as title (
9782:
9749:
9743:
9633:
9619:: CS1 maint: archived copy as title (
9581:
9567:: CS1 maint: archived copy as title (
9529:
9500:: CS1 maint: archived copy as title (
9462:
9448:: CS1 maint: archived copy as title (
9410:
9384:: CS1 maint: archived copy as title (
9346:
9228:: CS1 maint: archived copy as title (
9189:
9175:: CS1 maint: archived copy as title (
9137:
9124:
9054:Zeitschrift für Mathematik und Physik
8941:: CS1 maint: archived copy as title (
8903:
8889:: CS1 maint: archived copy as title (
8851:
8699:is primitive is sometimes said to be
7335:{\displaystyle \scriptstyle \lambda }
7178:{\displaystyle \scriptstyle \lambda }
6932:. The third fact is also elementary:
6804:such that there exists an eigenvalue
6745:—by the Perron–Frobenius eigenvector
6369:| = 1), so it tends to infinity when
5141:with the asterisks zeroised. If each
3938:
3144:is invariant under multiplication by
2708:{\displaystyle r\in \mathbb {R} ^{+}}
10108:
10016:Langville, Amy; Meyer, Carl (2006),
9283:
9277:
7792:
6793:for the Perron–Frobenius eigenvalue
6684:be its Perron–Frobenius eigenvalue.
6645:for the Perron-Frobenius eigenvalue
6436:Assuming there exists an eigenpair (
5950:If there is a Jordan cell, then the
5896:are both real, so the null space of
5693:as any non-negative eigenvector for
5294:
5103:can also be studied. The inverse of
4278:is the characteristic polynomial of
2046:does not have non-trivial invariant
813:be strictly positive vectors. Then,
191:
9104:Linear Algebra and Its Applications
8010:eigenvalues on the unit circle. If
7883:) = 1 then it can be decomposed as
7835:is a positive right eigenvector of
7776:Another way to argue it is via the
7646:applied to the non-negative matrix
6586:is the Perron–Frobenius eigenvalue
5856:
3719:is the Perron–Frobenius eigenvalue.
2236:, and there is an edge from vertex
1145:a strictly positive vector. Then,
792:is the Perron–Frobenius eigenvalue.
788:over all strictly positive vectors
746:is the Perron–Frobenius eigenvalue.
47:with positive entries has a unique
13:
10268:, Cambridge University Press, 1990
10218:
9401:, p. section XIII.5 theorem 9
8831:10.1093/oxfordjournals.cje.a035479
8191:
7614:
7507:
6417:No other non-negative eigenvectors
6123:
6056:
6030:
6001:
5888:can be chosen to be real, because
5605:This can be established using the
5244:The theorem has particular use in
4328:
4226:is the greatest common divisor of
1380:
1362:
956:
938:
884:
866:
607:
14:
10390:
10273:Nonlinear Perron-Frobenius Theory
8476:
8390:
8298:
8212:
8026:is non-negative and irreducible,
7787:
6582:is a real-valued function, whose
5315:Ruelle–Perron–Frobenius operators
5121:isn't invertible then neither is
5070:is a permutation matrix and each
4886:
4859:normal form of a reducible matrix
4629:
4559:can be, depending on the size of
4009:for some diagonal unitary matrix
3848:
2447:{\displaystyle t\mapsto \exp(tA)}
1843:
1778:
10271:Bas Lemmens and Roger Nussbaum,
10113:, John Wiley&Sons,New York,
9998:The Theory of Matrices, Volume 2
9247:chapter XIII.3 theorem 3 page 66
8180:may be computed as the limit of
8176:which eventually decay to zero.
7761:≥ the sum of the numbers in row
7364:is a corresponding eigenvector,
7088:its Perron–Frobenius eigenvalue
5908:by −1). Given maximal possible
5904:is positive (otherwise multiply
5340:
3715:is a real valued function whose
3602:
3569:
3539:
2838:
2816:
2742:The Perron–Frobenius eigenvalue
2406:{\displaystyle \mathbb {C} ^{n}}
2377:{\displaystyle (\mathbb {C} ,+)}
2343:{\displaystyle \mathbb {R} ^{n}}
2314:{\displaystyle (\mathbb {R} ,+)}
2212:One can associate with a matrix
1695:The left and right eigenvectors
784:is a real valued function whose
738:is a real valued function whose
683:. This projection is called the
342:There is a positive real number
91:); to Internet search engines (
10353:(2nd ed.), Springer-Verlag
10284:London: Springer-Verlag, 1993.
9880:
9828:
9787:chapter 8 claim 8.2.10 page 666
9776:
9728:
9703:
9675:
9639:
9627:
9575:
9523:
9508:
9456:
9404:
9392:
9305:
9236:
9183:
9091:
9082:
8172:representing the transients of
4839:
3955:) is a positive matrix. (Meyer
3579:{\displaystyle f(\mathbf {x} )}
3124:, consequently the spectrum of
3060:{\displaystyle \omega =2\pi /h}
2592:be an irreducible non-negative
331:{\displaystyle 1\leq i,j\leq n}
10278:S. P. Meyn and R. L. Tweedie,
10020:, Princeton University Press,
9073:
9064:
9041:
9022:
8972:. Princeton University Press.
8819:Cambridge Journal of Economics
8810:
8658:
7815:is its Perron projection then
7736:and the smallest component of
7631:
7623:
7577:
7559:
7502:
7496:
7467:
7459:
7451:
7443:
7433:
7422:
7414:
7406:
7396:
7385:
7357:{\displaystyle \scriptstyle x}
7305:
7297:
7250:
7232:
7202:
7194:
6665:Perron projection as a limit:
6168:
6155:
6096:
6084:
6071:
4422:
4406:
4325:
3990:). If equality holds (i.e. if
3607:
3595:
3573:
3565:
2657:
2651:
2476:if it is non-negative and its
2441:
2432:
2423:
2371:
2357:
2308:
2294:
2094:its image under the action of
1225:
1215:
677:projection onto the eigenspace
604:
417:
411:
225:
209:
1:
10264:R. A. Horn and C.R. Johnson,
9900:
9714:. New York: Springer-Verlag.
9689:. Boca Raton, FL: CRC Press.
8054:. The spectral projection of
7879:is a primitive matrix with ρ(
7797:The proof now proceeds using
6966:, so taking the limit yields
6601:. The proof requires to show
3117:{\displaystyle e^{i\omega }A}
2058:of standard basis vectors of
9685:; Cvetkovic, Dragos (2009).
9117:10.1016/0024-3795(85)90237-X
8140: ⊕ (1 −
8050:represents the harmonics of
7993:
7084:For any non-negative matrix
7045:. Hence its trace equals to
6540:, one defines the function
5602:respectively non-negative.)
3546:{\displaystyle \mathbf {x} }
3167:{\displaystyle e^{i\omega }}
2845:{\displaystyle \mathbf {w} }
2823:{\displaystyle \mathbf {v} }
1886:, the maximum eigenvalue is
1141:be a probability vector and
563:Perron-Frobenius eigenvector
503:such that all components of
472:There exists an eigenvector
102:
7:
10317:Encyclopedia of Mathematics
10018:Google page rank and beyond
9710:Mackey, Michael C. (1992).
9661:. Cambridge: Cambridge UP.
9657:Combinatorial Matrix Theory
8743:
8718:Perron–Frobenius eigenvalue
7998:Suppose in addition that ρ(
7516:
5479:. Because of the choice of
5408:Proof for positive matrices
5347:Brouwer fixed point theorem
5085:, moreover the spectrum of
4013:(i.e. diagonal elements of
3688:{\displaystyle x_{i}\neq 0}
2737:Perron–Frobenius eigenvalue
352:Perron–Frobenius eigenvalue
293:{\displaystyle a_{ij}>0}
10:
10395:
10374:Theorems in linear algebra
10312:"Perron–Frobenius theorem"
10310:Suprunenko, D.A. (2001) ,
10001:, AMS Chelsea Publishing,
9912:"Zur Theorie der Matrices"
9771:10.3318/PRIA.2002.102.1.29
8956:Langville & Meyer 2006
8710:and is sometimes called a
8148:so the difference between
7974:) = 1. So we consider the
7092:satisfies the inequality:
5420:) = 1 (otherwise consider
5298:
4587:is not positive for every
4449:= 1. Moreover, the matrix
3634:{\displaystyle _{i}/x_{i}}
2669:{\displaystyle \rho (A)=r}
2465:if it is not irreducible.
2456:irreducible representation
2006:Classification of matrices
1971:for some positive integer
761:) be the maximum value of
708:) be the minimum value of
671:= 1. Moreover, the matrix
231:{\displaystyle A=(a_{ij})}
10351:Matrix Iterative Analysis
10182:Mathematische Zeitschrift
10154:Mathematische Zeitschrift
10026:10.1007/s10791-008-9063-y
7276:satisfies the inequality
7267:Gershgorin circle theorem
6767:are normalized such that
5613:the sequence of vectors
5323:thermodynamic equilibrium
5148:is invertible then so is
4021:, non-diagonal are zero).
2611:{\displaystyle N\times N}
1821:, the maximum eigenvalue
447:characteristic polynomial
386:is strictly smaller than
374:and any other eigenvalue
257:{\displaystyle n\times n}
9739:section XIII.2.2 page 54
9636:, p. 2.43 (page 51)
9519:section XIII.2.2 page 62
9258:Kitchens, Bruce (1998),
8804:
8708:probability distribution
8695:such that 1 +
7053:acts identically on the
6839:on the diagonal (denote
6552:is the minimum value of
6532:Collatz–Wielandt formula
6452:is positive, and given (
5876:and another eigenvector
5532:
5273:strongly connected graph
4461:, the Perron projection.
3966:Wielandt's theorem. If |
3927:need not be square, and
3587:be the minimum value of
1963:ranges over the complex
1914:{\displaystyle \omega r}
1740:{\displaystyle w^{T}v=1}
445:is a simple root of the
423:{\displaystyle \rho (A)}
89:DeGroot learning process
87:); to social networks (
73:subshifts of finite type
25:Perron–Frobenius theorem
16:Theory in linear algebra
9048:Landau, Edmund (1915),
8034:, and the cyclic group
7653:Another inequality is:
7025:, because the image of
6890:(without assuming that
6858:will be conjugate (1, (
6781:is a positive operator.
5924:is not maximum. Vector
5561:,... are all positive.
5289:subshift of finite type
5164:= 0 the inverse of 1 −
4599: + 2, since (
4442:are normalized so that
3187:{\displaystyle \omega }
2510:of all natural numbers
2508:greatest common divisor
2232:vertices labeled 1,...,
1956:{\displaystyle \omega }
1705:stochastic eigenvectors
667:are normalized so that
146:exponential growth rate
81:Hawkins–Simon condition
10252:Algebraic Graph Theory
9031:Deutsches Wochenschach
8756:Z-matrix (mathematics)
8734:index of imprimitivity
8712:stochastic eigenvector
8624:
8454:
8368:
8276:
8164:= (1 −
8122:
8093:
7799:spectral decomposition
7711:
7640:
7589:
7557:
7476:
7358:
7336:
7314:
7259:
7179:
7150:
6718:, which commutes with
6468:(i.e. eigenvector for
6341:
6175:
5958:tends to infinity for
5649:. (The initial vector
5541:, assume there exists
5246:algebraic graph theory
5240:Algebraic graph theory
5051:
4830:
4432:
4209:
3959:). For a non-negative
3912:
3819:
3709:
3689:
3656:
3635:
3580:
3547:
3514:
3487:
3237:
3217:
3216:{\displaystyle h>1}
3188:
3168:
3138:
3118:
3081:
3061:
3017:
2997:
2977:
2953:
2933:
2913:
2890:
2866:
2846:
2824:
2802:
2780:
2756:
2729:
2709:
2670:
2632:
2612:
2586:
2448:
2407:
2378:
2344:
2315:
2189:
1994:
1957:
1936:
1915:
1880:
1815:
1741:
1675:
1578:
1550:
1491:
1256:
1204:
1115:
1087:
1028:
679:corresponding to
657:
424:
332:
294:
258:
232:
117:respectively describe
10109:Minc, Henryk (1988),
9917:Mathematische Annalen
9750:Smith, Roger (2006),
8625:
8455:
8369:
8277:
8123:
8079:
7976:peripheral projection
7950:Peripheral projection
7712:
7641:
7590:
7537:
7477:
7359:
7337:
7315:
7260:
7180:
7151:
6688:There exists a limit
6567:taken over all those
6342:
6176:
5537:Given a non-negative
5188:are both invertible.
5099:The invertibility of
5052:
4849:Non-negative matrices
4831:
4543:is positive for some
4433:
4210:
3931:need not divide
3913:
3820:
3710:
3690:
3657:
3642:taken over all those
3636:
3581:
3548:
3515:
3488:
3238:
3218:
3189:
3169:
3139:
3119:
3082:
3062:
3018:
2998:
2978:
2954:
2934:
2914:
2891:
2867:
2847:
2825:
2803:
2781:
2757:
2730:
2710:
2671:
2633:
2613:
2587:
2484:(i.e. all entries of
2449:
2408:
2379:
2345:
2316:
2253:≠ 0. Then the matrix
2190:
1995:
1958:
1937:
1916:
1881:
1816:
1751:Non-negative matrices
1742:
1676:
1579:
1530:
1465:
1257:
1184:
1116:
1067:
1002:
723:taken over all those
658:
571:principal eigenvector
425:
333:
295:
259:
233:
148:of the matrix powers
10111:Nonnegative matrices
10078:Meyer, Carl (2000),
9534:example 8.3.3 p. 678
9467:example 8.3.2 p. 677
9289:Nonnegative matrices
9194:chapter 8.3 page 670
8798:Krein–Rutman theorem
8783:Quasipositive matrix
8730:spectral idempotents
8468:
8382:
8290:
8204:
8062:
7660:
7599:
7490:
7368:
7346:
7324:
7280:
7189:
7167:
7099:
6894:is diagonalizable).
6629:. The definition of
6512:> 0, so one has:
6206:
5987:
5475:is an eigenvalue of
5301:Krein–Rutman theorem
5279:Finite Markov chains
4878:
4615:
4551:is positive for all
4314:
4036:
3834:
3734:
3699:
3666:
3646:
3592:
3559:
3535:
3504:
3253:
3227:
3201:
3178:
3148:
3128:
3095:
3071:
3034:
3007:
2987:
2967:
2943:
2923:
2903:
2880:
2856:
2834:
2812:
2792:
2786:are one-dimensional.
2770:
2746:
2719:
2684:
2645:
2622:
2596:
2576:
2417:
2388:
2354:
2325:
2291:
2285:group representation
2121:
1984:
1947:
1926:
1902:
1829:
1764:
1715:
1590:
1274:
1149:
817:
593:
575:dominant eigenvector
453:. Consequently, the
405:
370:is an eigenvalue of
360:principal eigenvalue
304:
268:
242:
200:
67:); to the theory of
57:nonnegative matrices
9683:Brualdi, Richard A.
9647:Brualdi, Richard A.
9351:claim 8.3.11 p. 675
8726:spectral projectors
8722:dominant eigenvalue
7831:so every column of
7320:for any eigenvalue
7274:matrix induced norm
7163:with an eigenvalue
6819:is diagonalizable,
6791:spectral projection
6708:projection operator
6448:, such that vector
6188:is bounded for all
5920:is zero, otherwise
5641:| converges to the
5213:Stochastic matrices
4455:spectral projection
4032:, it is true that:
2618:matrix with period
2041:Definition 1 :
2022:square matrix over
1711:sums to one, while
567:leading eigenvector
465:, the transpose of
364:dominant eigenvalue
182:Georg Frobenius
144:of the matrix. The
37:Georg Frobenius
10226:Robert J. Plemmons
10203:10338.dmlcz/100322
10194:10.1007/BF02230720
10166:10.1007/BF01180013
10142:10.24033/bsmf.1206
9939:10338.dmlcz/104432
9930:10.1007/BF01449896
9892:Richard A. Brualdi
9839:chapter 8 page 666
9142:chapter 8 page 665
8738:order of cyclicity
8634:are both equal to
8620:
8614:
8613:
8450:
8444:
8443:
8364:
8358:
8357:
8272:
8266:
8265:
8118:
8117:
7954:The analysis when
7707:
7682:
7672:
7636:
7635:
7585:
7584:
7536:
7472:
7471:
7354:
7353:
7332:
7331:
7310:
7309:
7255:
7254:
7230:
7220:
7175:
7174:
7146:
7129:
7119:
6741:(respectively for
6337:
6328:
6253:
6171:
6154:
6070:
5549:is positive, then
5335:point-set topology
5311:transfer operators
5047:
5041:
5040:
4826:
4820:
4819:
4428:
4377:
4332:
4205:
4199:
3994:is eigenvalue for
3957:claim 8.3.5 p. 672
3939:Further properties
3908:
3902:
3901:
3815:
3798:
3788:
3756:
3746:
3705:
3685:
3652:
3631:
3576:
3543:
3510:
3483:
3474:
3233:
3213:
3184:
3164:
3134:
3114:
3077:
3067:. Then the matrix
3057:
3013:
2993:
2973:
2949:
2929:
2909:
2886:
2862:
2852:, with eigenvalue
2842:
2820:
2798:
2776:
2752:
2725:
2705:
2666:
2628:
2608:
2582:
2444:
2403:
2374:
2340:
2311:
2268:strongly connected
2185:
2176:
2109:permutation matrix
1990:
1953:
1932:
1911:
1876:
1870:
1869:
1811:
1805:
1804:
1737:
1686:primitive matrices
1671:
1654:
1644:
1612:
1602:
1574:
1464:
1409:
1353:
1298:
1252:
1183:
1167:
1111:
1001:
985:
929:
913:
857:
841:
653:
611:
559:Perron eigenvector
420:
356:leading eigenvalue
328:
290:
254:
228:
83:); to demography (
45:real square matrix
43:), asserts that a
10347:Varga, Richard S.
10305:978-0-387-29765-1
10254:, Springer, 2001.
10094:978-0-89871-454-8
10035:978-0-691-12202-1
10008:978-0-8218-2664-5
9993:Gantmacher, Felix
9888:Olga Taussky-Todd
9721:978-0-387-97702-7
9696:978-1-4200-8223-4
9668:978-0-521-32265-2
9651:Ryser, Herbert J.
9325:978-0-12-384934-2
8964:Langville, Amy N.
8789:Positive operator
8675:means > 0 and
8665:strictly positive
7903:+ (1 −
7887:⊕ (1 −
7793:Perron projection
7673:
7663:
7515:
7221:
7211:
7120:
7110:
7057:-eigenvector for
7021:are multiples of
7005:-eigenvector for
6657:, which implies
6145:
6104:
6101:
6095:
6092:
6061:
5872:corresponding to
5645:with the maximum
5501:Gelfand's formula
5400:is the period of
5307:compact operators
5295:Compact operators
5219:stochastic matrix
5137:, in other words
5129:. Conversely let
4583:below, the power
4457:corresponding to
4344:
4317:
3789:
3779:
3747:
3737:
3708:{\displaystyle f}
3655:{\displaystyle i}
3528:–Wielandt formula
3513:{\displaystyle O}
3236:{\displaystyle P}
3137:{\displaystyle A}
3080:{\displaystyle A}
3016:{\displaystyle h}
2996:{\displaystyle r}
2976:{\displaystyle r}
2952:{\displaystyle h}
2932:{\displaystyle h}
2912:{\displaystyle A}
2889:{\displaystyle r}
2865:{\displaystyle r}
2801:{\displaystyle A}
2779:{\displaystyle r}
2755:{\displaystyle r}
2728:{\displaystyle A}
2631:{\displaystyle h}
2585:{\displaystyle A}
1993:{\displaystyle r}
1935:{\displaystyle r}
1645:
1635:
1603:
1593:
1447:
1432:
1413:
1412:
1394:
1389:
1336:
1321:
1302:
1301:
1283:
1168:
1158:
986:
970:
965:
914:
898:
893:
842:
826:
694:–Wielandt formula
685:Perron projection
596:
354:(also called the
264:positive matrix:
192:Positive matrices
160:with the largest
140:that make up the
121:with exclusively
77:Okishio's theorem
75:); to economics (
69:dynamical systems
10386:
10379:Markov processes
10354:
10325:(The claim that
10324:
10224:Abraham Berman,
10214:
10205:
10176:
10145:
10144:
10123:
10105:
10103:
10097:, archived from
10086:
10074:
10046:
10011:
9988:
9981:Frobenius, Georg
9976:
9969:Frobenius, Georg
9964:
9955:Frobenius, Georg
9950:
9941:
9895:
9884:
9878:
9877:
9871:
9863:
9861:
9860:
9855:on March 7, 2010
9854:
9848:. Archived from
9847:
9832:
9826:
9825:
9819:
9811:
9809:
9808:
9803:on March 7, 2010
9802:
9796:. Archived from
9795:
9780:
9774:
9773:
9756:
9747:
9741:
9732:
9726:
9725:
9707:
9701:
9700:
9679:
9673:
9672:
9660:
9643:
9637:
9631:
9625:
9624:
9618:
9610:
9608:
9607:
9602:on March 7, 2010
9601:
9595:. Archived from
9594:
9579:
9573:
9572:
9566:
9558:
9556:
9555:
9550:on March 7, 2010
9549:
9543:. Archived from
9542:
9527:
9521:
9512:
9506:
9505:
9499:
9491:
9489:
9488:
9483:on March 7, 2010
9482:
9476:. Archived from
9475:
9460:
9454:
9453:
9447:
9439:
9437:
9436:
9431:on March 7, 2010
9430:
9424:. Archived from
9423:
9408:
9402:
9396:
9390:
9389:
9383:
9375:
9373:
9372:
9367:on March 7, 2010
9366:
9360:. Archived from
9359:
9344:
9338:
9337:
9309:
9303:
9302:
9281:
9275:
9274:
9255:
9249:
9240:
9234:
9233:
9227:
9219:
9217:
9216:
9211:on March 7, 2010
9210:
9204:. Archived from
9203:
9187:
9181:
9180:
9174:
9166:
9164:
9163:
9158:on March 7, 2010
9157:
9151:. Archived from
9150:
9135:
9122:
9121:
9119:
9095:
9089:
9086:
9080:
9077:
9071:
9068:
9062:
9061:
9045:
9039:
9038:
9026:
9020:
9011:
9005:
9004:
8998:
8990:
8988:
8987:
8953:
8947:
8946:
8940:
8932:
8930:
8929:
8924:on March 7, 2010
8923:
8917:. Archived from
8916:
8901:
8895:
8894:
8888:
8880:
8878:
8877:
8872:on March 7, 2010
8871:
8865:. Archived from
8864:
8849:
8843:
8842:
8814:
8794:
8629:
8627:
8626:
8621:
8619:
8615:
8459:
8457:
8456:
8451:
8449:
8445:
8373:
8371:
8370:
8365:
8363:
8359:
8281:
8279:
8278:
8273:
8271:
8267:
8188: → ∞.
8127:
8125:
8124:
8119:
8116:
8115:
8106:
8105:
8092:
8087:
8078:
8077:
7927: → ∞.
7919:as the limit of
7716:
7714:
7713:
7708:
7695:
7694:
7681:
7671:
7645:
7643:
7642:
7637:
7634:
7626:
7618:
7617:
7594:
7592:
7591:
7586:
7580:
7575:
7574:
7562:
7556:
7551:
7535:
7511:
7510:
7505:
7481:
7479:
7478:
7473:
7470:
7462:
7454:
7446:
7441:
7436:
7425:
7417:
7409:
7404:
7399:
7388:
7363:
7361:
7360:
7355:
7341:
7339:
7338:
7333:
7319:
7317:
7316:
7311:
7308:
7300:
7264:
7262:
7261:
7256:
7253:
7248:
7247:
7235:
7229:
7219:
7205:
7197:
7185:it is true that
7184:
7182:
7181:
7176:
7155:
7153:
7152:
7147:
7142:
7141:
7128:
7118:
6990:-eigenspace for
6346:
6344:
6343:
6338:
6333:
6332:
6325:
6324:
6306:
6305:
6285:
6284:
6264:
6263:
6258:
6257:
6218:
6217:
6180:
6178:
6177:
6172:
6167:
6166:
6153:
6144:
6127:
6126:
6117:
6116:
6102:
6099:
6093:
6090:
6083:
6082:
6069:
6060:
6059:
6050:
6049:
6034:
6033:
6021:
6020:
6005:
6004:
5940:does not exist.
5930:previous section
5857:Multiplicity one
5327:dynamical system
5275:is irreducible.
5269:adjacency matrix
5056:
5054:
5053:
5048:
5046:
5042:
5037:
5036:
4972:
4938:
4937:
4899:
4898:
4835:
4833:
4832:
4827:
4825:
4821:
4748:
4437:
4435:
4434:
4429:
4421:
4420:
4402:
4401:
4392:
4387:
4386:
4376:
4340:
4331:
4214:
4212:
4211:
4206:
4204:
4203:
4196:
4195:
4158:
4124:
4123:
4085:
4084:
4064:
4063:
4051:
4050:
3917:
3915:
3914:
3909:
3907:
3903:
3824:
3822:
3821:
3816:
3811:
3810:
3797:
3787:
3769:
3768:
3755:
3745:
3714:
3712:
3711:
3706:
3694:
3692:
3691:
3686:
3678:
3677:
3661:
3659:
3658:
3653:
3640:
3638:
3637:
3632:
3630:
3629:
3620:
3615:
3614:
3605:
3585:
3583:
3582:
3577:
3572:
3552:
3550:
3549:
3544:
3542:
3519:
3517:
3516:
3511:
3492:
3490:
3489:
3484:
3479:
3478:
3446:
3445:
3432:
3431:
3383:
3344:
3343:
3300:
3299:
3274:
3273:
3242:
3240:
3239:
3234:
3222:
3220:
3219:
3214:
3193:
3191:
3190:
3185:
3173:
3171:
3170:
3165:
3163:
3162:
3143:
3141:
3140:
3135:
3123:
3121:
3120:
3115:
3110:
3109:
3086:
3084:
3083:
3078:
3066:
3064:
3063:
3058:
3053:
3022:
3020:
3019:
3014:
3002:
3000:
2999:
2994:
2982:
2980:
2979:
2974:
2958:
2956:
2955:
2950:
2938:
2936:
2935:
2930:
2918:
2916:
2915:
2910:
2895:
2893:
2892:
2887:
2871:
2869:
2868:
2863:
2851:
2849:
2848:
2843:
2841:
2829:
2827:
2826:
2821:
2819:
2807:
2805:
2804:
2799:
2785:
2783:
2782:
2777:
2761:
2759:
2758:
2753:
2734:
2732:
2731:
2726:
2714:
2712:
2711:
2706:
2704:
2703:
2698:
2675:
2673:
2672:
2667:
2637:
2635:
2634:
2629:
2617:
2615:
2614:
2609:
2591:
2589:
2588:
2583:
2501:period of index
2453:
2451:
2450:
2445:
2412:
2410:
2409:
2404:
2402:
2401:
2396:
2383:
2381:
2380:
2375:
2364:
2349:
2347:
2346:
2341:
2339:
2338:
2333:
2320:
2318:
2317:
2312:
2301:
2194:
2192:
2191:
2186:
2181:
2180:
2142:
2141:
2090: <
1999:
1997:
1996:
1991:
1962:
1960:
1959:
1954:
1941:
1939:
1938:
1933:
1920:
1918:
1917:
1912:
1885:
1883:
1882:
1877:
1875:
1871:
1820:
1818:
1817:
1812:
1810:
1806:
1746:
1744:
1743:
1738:
1727:
1726:
1680:
1678:
1677:
1672:
1667:
1666:
1653:
1643:
1625:
1624:
1611:
1601:
1583:
1581:
1580:
1575:
1570:
1569:
1560:
1559:
1549:
1544:
1529:
1524:
1523:
1514:
1513:
1501:
1500:
1490:
1485:
1463:
1445:
1430:
1410:
1408:
1390:
1388:
1384:
1383:
1373:
1366:
1365:
1355:
1352:
1334:
1319:
1299:
1297:
1261:
1259:
1258:
1253:
1248:
1247:
1238:
1233:
1232:
1214:
1213:
1203:
1198:
1182:
1166:
1120:
1118:
1117:
1112:
1107:
1106:
1097:
1096:
1086:
1081:
1066:
1061:
1060:
1051:
1050:
1038:
1037:
1027:
1022:
1000:
984:
966:
964:
960:
959:
949:
942:
941:
931:
928:
912:
894:
892:
888:
887:
877:
870:
869:
859:
856:
840:
662:
660:
659:
654:
652:
651:
636:
635:
626:
621:
620:
610:
499:with eigenvalue
429:
427:
426:
421:
337:
335:
334:
329:
299:
297:
296:
291:
283:
282:
263:
261:
260:
255:
237:
235:
234:
229:
224:
223:
174:Oskar Perron
29:Oskar Perron
10394:
10393:
10389:
10388:
10387:
10385:
10384:
10383:
10359:
10358:
10333:
10266:Matrix Analysis
10221:
10219:Further reading
10150:Collatz, Lothar
10121:
10101:
10095:
10084:
10064:10.1137/1035004
10036:
10009:
9903:
9898:
9885:
9881:
9865:
9864:
9858:
9856:
9852:
9845:
9843:"Archived copy"
9841:
9833:
9829:
9813:
9812:
9806:
9804:
9800:
9793:
9791:"Archived copy"
9789:
9781:
9777:
9754:
9748:
9744:
9735:Gantmacher 2000
9733:
9729:
9722:
9708:
9704:
9697:
9680:
9676:
9669:
9644:
9640:
9632:
9628:
9612:
9611:
9605:
9603:
9599:
9592:
9590:"Archived copy"
9588:
9580:
9576:
9560:
9559:
9553:
9551:
9547:
9540:
9538:"Archived copy"
9536:
9528:
9524:
9515:Gantmacher 2000
9513:
9509:
9493:
9492:
9486:
9484:
9480:
9473:
9471:"Archived copy"
9469:
9461:
9457:
9441:
9440:
9434:
9432:
9428:
9421:
9419:"Archived copy"
9417:
9409:
9405:
9399:Gantmacher 2000
9397:
9393:
9377:
9376:
9370:
9368:
9364:
9357:
9355:"Archived copy"
9353:
9345:
9341:
9326:
9310:
9306:
9299:
9282:
9278:
9272:
9256:
9252:
9243:Gantmacher 2000
9241:
9237:
9221:
9220:
9214:
9212:
9208:
9201:
9199:"Archived copy"
9197:
9188:
9184:
9168:
9167:
9161:
9159:
9155:
9148:
9146:"Archived copy"
9144:
9136:
9125:
9096:
9092:
9087:
9083:
9078:
9074:
9069:
9065:
9046:
9042:
9027:
9023:
9012:
9008:
8992:
8991:
8985:
8983:
8980:
8954:
8950:
8934:
8933:
8927:
8925:
8921:
8914:
8912:"Archived copy"
8910:
8902:
8898:
8882:
8881:
8875:
8873:
8869:
8862:
8860:"Archived copy"
8858:
8850:
8846:
8815:
8811:
8807:
8792:
8750:Min-max theorem
8746:
8681:decomposability
8661:
8612:
8611:
8606:
8601:
8596:
8591:
8585:
8584:
8579:
8574:
8569:
8564:
8558:
8557:
8552:
8547:
8542:
8537:
8531:
8530:
8525:
8520:
8515:
8510:
8504:
8503:
8498:
8493:
8488:
8483:
8475:
8471:
8469:
8466:
8465:
8442:
8441:
8436:
8431:
8425:
8424:
8419:
8414:
8408:
8407:
8402:
8397:
8389:
8385:
8383:
8380:
8379:
8356:
8355:
8350:
8345:
8333:
8332:
8327:
8322:
8316:
8315:
8310:
8305:
8297:
8293:
8291:
8288:
8287:
8264:
8263:
8258:
8253:
8247:
8246:
8241:
8236:
8230:
8229:
8224:
8219:
8211:
8207:
8205:
8202:
8201:
8194:
8192:Counterexamples
8111:
8107:
8098:
8094:
8088:
8083:
8070:
8066:
8063:
8060:
8059:
7996:
7952:
7795:
7790:
7760:
7745:
7687:
7683:
7677:
7667:
7661:
7658:
7657:
7630:
7622:
7613:
7609:
7600:
7597:
7596:
7576:
7567:
7563:
7558:
7552:
7541:
7519:
7506:
7495:
7494:
7491:
7488:
7487:
7466:
7458:
7450:
7442:
7437:
7432:
7421:
7413:
7405:
7400:
7395:
7384:
7369:
7366:
7365:
7347:
7344:
7343:
7325:
7322:
7321:
7304:
7296:
7281:
7278:
7277:
7249:
7240:
7236:
7231:
7225:
7215:
7201:
7193:
7190:
7187:
7186:
7168:
7165:
7164:
7134:
7130:
7124:
7114:
7100:
7097:
7096:
7082:
7068:) = 1, implies
6982:), so image of
6877:
6864:
6845:
6838:
6829:
6696:, denote it by
6674:
6576:
6566:
6557:
6534:
6419:
6373:does so. Since
6356:
6327:
6326:
6320:
6316:
6314:
6308:
6307:
6295:
6291:
6286:
6280:
6276:
6269:
6268:
6259:
6252:
6251:
6246:
6240:
6239:
6234:
6224:
6223:
6222:
6213:
6209:
6207:
6204:
6203:
6198:
6187:
6162:
6158:
6149:
6140:
6122:
6118:
6112:
6108:
6078:
6074:
6065:
6055:
6051:
6045:
6041:
6029:
6025:
6016:
6012:
6000:
5996:
5988:
5985:
5984:
5957:
5859:
5852:
5843:
5835:
5825:
5812:
5801:>0, hence:
5800:
5782:
5774:
5764:
5752:
5741:>0, hence:
5740:
5718:
5668:
5662:
5655:
5640:
5631:
5622:
5588:
5535:
5410:
5379:spectral radius
5363:
5355:spectral theory
5343:
5313:, or sometimes
5303:
5297:
5281:
5265:
5260:if and only if
5242:
5217:A row (column)
5215:
5207:
5200:
5146:
5119:
5112:
5094:
5075:
5039:
5038:
5032:
5028:
5026:
5021:
5016:
5011:
5005:
5004:
4999:
4994:
4989:
4984:
4978:
4977:
4971:
4966:
4961:
4955:
4954:
4949:
4944:
4939:
4933:
4929:
4927:
4921:
4920:
4915:
4910:
4905:
4900:
4894:
4890:
4885:
4881:
4879:
4876:
4875:
4851:
4842:
4818:
4817:
4812:
4807:
4802:
4797:
4792:
4786:
4785:
4780:
4775:
4770:
4765:
4760:
4754:
4753:
4747:
4742:
4737:
4732:
4726:
4725:
4720:
4715:
4710:
4705:
4700:
4694:
4693:
4688:
4683:
4678:
4673:
4668:
4662:
4661:
4656:
4651:
4646:
4641:
4636:
4628:
4624:
4616:
4613:
4612:
4606:
4529:
4525:
4515:
4506:
4416:
4412:
4397:
4393:
4388:
4382:
4378:
4348:
4336:
4321:
4315:
4312:
4311:
4299:
4295:
4291:
4275:
4274:
4268:
4267:
4261:
4260:
4220:
4198:
4197:
4191:
4187:
4185:
4180:
4175:
4170:
4164:
4163:
4157:
4152:
4147:
4141:
4140:
4135:
4130:
4125:
4119:
4115:
4113:
4107:
4106:
4101:
4096:
4091:
4086:
4080:
4076:
4069:
4068:
4056:
4052:
4046:
4042:
4037:
4034:
4033:
3941:
3926:
3900:
3899:
3894:
3889:
3883:
3882:
3877:
3872:
3866:
3865:
3860:
3855:
3847:
3843:
3835:
3832:
3831:
3803:
3799:
3793:
3783:
3761:
3757:
3751:
3741:
3735:
3732:
3731:
3700:
3697:
3696:
3673:
3669:
3667:
3664:
3663:
3647:
3644:
3643:
3625:
3621:
3616:
3610:
3606:
3601:
3593:
3590:
3589:
3568:
3560:
3557:
3556:
3538:
3536:
3533:
3532:
3505:
3502:
3501:
3473:
3472:
3467:
3462:
3457:
3452:
3447:
3441:
3437:
3434:
3433:
3421:
3417:
3415:
3410:
3405:
3400:
3395:
3389:
3388:
3382:
3377:
3372:
3367:
3361:
3360:
3355:
3350:
3345:
3339:
3335:
3333:
3328:
3322:
3321:
3316:
3311:
3306:
3301:
3295:
3291:
3289:
3279:
3278:
3266:
3262:
3254:
3251:
3250:
3228:
3225:
3224:
3202:
3199:
3198:
3179:
3176:
3175:
3155:
3151:
3149:
3146:
3145:
3129:
3126:
3125:
3102:
3098:
3096:
3093:
3092:
3072:
3069:
3068:
3049:
3035:
3032:
3031:
3008:
3005:
3004:
2988:
2985:
2984:
2968:
2965:
2964:
2944:
2941:
2940:
2924:
2921:
2920:
2904:
2901:
2900:
2881:
2878:
2877:
2857:
2854:
2853:
2837:
2835:
2832:
2831:
2815:
2813:
2810:
2809:
2793:
2790:
2789:
2771:
2768:
2767:
2747:
2744:
2743:
2735:. It is called
2720:
2717:
2716:
2699:
2694:
2693:
2685:
2682:
2681:
2646:
2643:
2642:
2640:spectral radius
2623:
2620:
2619:
2597:
2594:
2593:
2577:
2574:
2573:
2570:
2548:
2523:
2499:and define the
2488:are positive).
2418:
2415:
2414:
2397:
2392:
2391:
2389:
2386:
2385:
2360:
2355:
2352:
2351:
2334:
2329:
2328:
2326:
2323:
2322:
2297:
2292:
2289:
2288:
2265:
2252:
2244:precisely when
2227:
2175:
2174:
2169:
2163:
2162:
2157:
2147:
2146:
2134:
2130:
2122:
2119:
2118:
2085:
2084:
2073:
2072:
2054:spanned by any
2052:linear subspace
2008:
1985:
1982:
1981:
1948:
1945:
1944:
1927:
1924:
1923:
1903:
1900:
1899:
1868:
1867:
1862:
1856:
1855:
1850:
1842:
1838:
1830:
1827:
1826:
1803:
1802:
1797:
1791:
1790:
1785:
1777:
1773:
1765:
1762:
1761:
1753:
1722:
1718:
1716:
1713:
1712:
1659:
1655:
1649:
1639:
1617:
1613:
1607:
1597:
1591:
1588:
1587:
1565:
1561:
1555:
1551:
1545:
1534:
1525:
1519:
1515:
1506:
1502:
1496:
1492:
1486:
1469:
1417:
1398:
1379:
1375:
1374:
1361:
1357:
1356:
1354:
1306:
1287:
1275:
1272:
1271:
1243:
1239:
1234:
1228:
1224:
1209:
1205:
1199:
1188:
1172:
1162:
1150:
1147:
1146:
1102:
1098:
1092:
1088:
1082:
1071:
1062:
1056:
1052:
1043:
1039:
1033:
1029:
1023:
1006:
990:
974:
955:
951:
950:
937:
933:
932:
930:
918:
902:
883:
879:
878:
865:
861:
860:
858:
846:
830:
818:
815:
814:
775:
766:
732:
722:
713:
647:
643:
631:
627:
622:
616:
612:
600:
594:
591:
590:
552:
524:> 0 for 1 ≤
523:
491:
482:
406:
403:
402:
400:spectral radius
305:
302:
301:
275:
271:
269:
266:
265:
243:
240:
239:
216:
212:
201:
198:
197:
194:
138:complex numbers
105:
17:
12:
11:
5:
10392:
10382:
10381:
10376:
10371:
10357:
10356:
10343:
10329:
10307:
10293:
10276:
10269:
10262:
10255:
10241:
10232:, 1994, SIAM.
10220:
10217:
10216:
10215:
10188:(1): 642–648,
10177:
10160:(1): 221–226,
10146:
10124:
10119:
10106:
10093:
10075:
10047:
10034:
10013:
10007:
9989:
9977:
9965:
9951:
9924:(2): 248–263,
9902:
9899:
9897:
9896:
9879:
9827:
9775:
9742:
9727:
9720:
9702:
9695:
9674:
9667:
9638:
9626:
9574:
9522:
9507:
9455:
9403:
9391:
9339:
9324:
9304:
9297:
9276:
9270:
9250:
9235:
9182:
9123:
9090:
9081:
9072:
9063:
9040:
9021:
9006:
8979:978-0691122021
8978:
8948:
8896:
8844:
8825:(2): 183–186.
8808:
8806:
8803:
8802:
8801:
8795:
8786:
8779:Metzler matrix
8776:
8773:Hurwitz matrix
8770:
8764:
8759:
8753:
8745:
8742:
8660:
8657:
8618:
8610:
8607:
8605:
8602:
8600:
8597:
8595:
8592:
8590:
8587:
8586:
8583:
8580:
8578:
8575:
8573:
8570:
8568:
8565:
8563:
8560:
8559:
8556:
8553:
8551:
8548:
8546:
8543:
8541:
8538:
8536:
8533:
8532:
8529:
8526:
8524:
8521:
8519:
8516:
8514:
8511:
8509:
8506:
8505:
8502:
8499:
8497:
8494:
8492:
8489:
8487:
8484:
8482:
8479:
8478:
8474:
8448:
8440:
8437:
8435:
8432:
8430:
8427:
8426:
8423:
8420:
8418:
8415:
8413:
8410:
8409:
8406:
8403:
8401:
8398:
8396:
8393:
8392:
8388:
8362:
8354:
8351:
8349:
8346:
8344:
8341:
8335:
8334:
8331:
8328:
8326:
8323:
8321:
8318:
8317:
8314:
8311:
8309:
8306:
8304:
8301:
8300:
8296:
8270:
8262:
8259:
8257:
8254:
8252:
8249:
8248:
8245:
8242:
8240:
8237:
8235:
8232:
8231:
8228:
8225:
8223:
8220:
8218:
8215:
8214:
8210:
8193:
8190:
8114:
8110:
8104:
8101:
8097:
8091:
8086:
8082:
8076:
8073:
8069:
7995:
7992:
7951:
7948:
7794:
7791:
7789:
7788:Further proofs
7786:
7756:
7743:
7718:
7717:
7706:
7703:
7699:
7693:
7690:
7686:
7680:
7676:
7670:
7666:
7633:
7629:
7625:
7621:
7616:
7612:
7608:
7605:
7583:
7579:
7573:
7570:
7566:
7561:
7555:
7550:
7547:
7544:
7540:
7534:
7531:
7528:
7525:
7522:
7518:
7514:
7509:
7504:
7501:
7498:
7469:
7465:
7461:
7457:
7453:
7449:
7445:
7440:
7435:
7431:
7428:
7424:
7420:
7416:
7412:
7408:
7403:
7398:
7394:
7391:
7387:
7383:
7380:
7377:
7374:
7352:
7330:
7307:
7303:
7299:
7295:
7292:
7289:
7286:
7252:
7246:
7243:
7239:
7234:
7228:
7224:
7218:
7214:
7209:
7204:
7200:
7196:
7173:
7157:
7156:
7145:
7140:
7137:
7133:
7127:
7123:
7117:
7113:
7108:
7104:
7081:
7078:
7017:). Columns of
6873:
6862:
6850:). The matrix
6843:
6834:
6827:
6783:
6782:
6775:
6754:
6731:
6701:
6673:
6663:
6574:
6562:
6553:
6533:
6530:
6418:
6415:
6354:
6348:
6347:
6336:
6331:
6323:
6319:
6315:
6313:
6310:
6309:
6304:
6301:
6298:
6294:
6290:
6287:
6283:
6279:
6275:
6274:
6272:
6267:
6262:
6256:
6250:
6247:
6245:
6242:
6241:
6238:
6235:
6233:
6230:
6229:
6227:
6221:
6216:
6212:
6196:
6185:
6170:
6165:
6161:
6157:
6152:
6148:
6143:
6139:
6136:
6133:
6130:
6125:
6121:
6115:
6111:
6107:
6098:
6089:
6086:
6081:
6077:
6073:
6068:
6064:
6058:
6054:
6048:
6044:
6040:
6037:
6032:
6028:
6024:
6019:
6015:
6011:
6008:
6003:
5999:
5995:
5992:
5955:
5858:
5855:
5848:
5839:
5831:
5821:
5808:
5796:
5778:
5770:
5760:
5748:
5736:
5714:
5699:
5698:
5666:
5660:
5653:
5636:
5627:
5617:
5587:
5584:
5579:will be zero.
5534:
5531:
5409:
5406:
5362:
5359:
5342:
5339:
5299:Main article:
5296:
5293:
5280:
5277:
5263:
5241:
5238:
5214:
5211:
5205:
5198:
5144:
5117:
5110:
5092:
5073:
5064:
5063:
5062:
5061:
5060:
5059:
5058:
5057:
5045:
5035:
5031:
5027:
5025:
5022:
5020:
5017:
5015:
5012:
5010:
5007:
5006:
5003:
5000:
4998:
4995:
4993:
4990:
4988:
4985:
4983:
4980:
4979:
4976:
4973:
4970:
4967:
4965:
4962:
4960:
4957:
4956:
4953:
4950:
4948:
4945:
4943:
4940:
4936:
4932:
4928:
4926:
4923:
4922:
4919:
4916:
4914:
4911:
4909:
4906:
4904:
4901:
4897:
4893:
4889:
4888:
4884:
4850:
4847:
4841:
4838:
4837:
4836:
4824:
4816:
4813:
4811:
4808:
4806:
4803:
4801:
4798:
4796:
4793:
4791:
4788:
4787:
4784:
4781:
4779:
4776:
4774:
4771:
4769:
4766:
4764:
4761:
4759:
4756:
4755:
4752:
4749:
4746:
4743:
4741:
4738:
4736:
4733:
4731:
4728:
4727:
4724:
4721:
4719:
4716:
4714:
4711:
4709:
4706:
4704:
4701:
4699:
4696:
4695:
4692:
4689:
4687:
4684:
4682:
4679:
4677:
4674:
4672:
4669:
4667:
4664:
4663:
4660:
4657:
4655:
4652:
4650:
4647:
4645:
4642:
4640:
4637:
4635:
4632:
4631:
4627:
4623:
4620:
4609:
4608:
4604:
4595: − 2
4533:
4532:
4527:
4523:
4511:
4502:
4488:
4477:
4476:) is positive.
4462:
4427:
4424:
4419:
4415:
4411:
4408:
4405:
4400:
4396:
4391:
4385:
4381:
4375:
4372:
4369:
4366:
4363:
4360:
4357:
4354:
4351:
4347:
4343:
4339:
4335:
4330:
4327:
4324:
4320:
4302:
4297:
4293:
4289:
4272:
4270:
4265:
4263:
4258:
4256:
4243:
4218:
4202:
4194:
4190:
4186:
4184:
4181:
4179:
4176:
4174:
4171:
4169:
4166:
4165:
4162:
4159:
4156:
4153:
4151:
4148:
4146:
4143:
4142:
4139:
4136:
4134:
4131:
4129:
4126:
4122:
4118:
4114:
4112:
4109:
4108:
4105:
4102:
4100:
4097:
4095:
4092:
4090:
4087:
4083:
4079:
4075:
4074:
4072:
4067:
4062:
4059:
4055:
4049:
4045:
4041:
4024:If some power
4022:
3964:
3940:
3937:
3922:
3906:
3898:
3895:
3893:
3890:
3888:
3885:
3884:
3881:
3878:
3876:
3873:
3871:
3868:
3867:
3864:
3861:
3859:
3856:
3854:
3851:
3850:
3846:
3842:
3839:
3828:
3827:
3826:
3825:
3814:
3809:
3806:
3802:
3796:
3792:
3786:
3782:
3778:
3775:
3772:
3767:
3764:
3760:
3754:
3750:
3744:
3740:
3726:
3725:
3721:
3720:
3704:
3684:
3681:
3676:
3672:
3651:
3628:
3624:
3619:
3613:
3609:
3604:
3600:
3597:
3575:
3571:
3567:
3564:
3541:
3509:
3498:
3497:
3496:
3495:
3493:
3482:
3477:
3471:
3468:
3466:
3463:
3461:
3458:
3456:
3453:
3451:
3448:
3444:
3440:
3436:
3435:
3430:
3427:
3424:
3420:
3416:
3414:
3411:
3409:
3406:
3404:
3401:
3399:
3396:
3394:
3391:
3390:
3387:
3384:
3381:
3378:
3376:
3373:
3371:
3368:
3366:
3363:
3362:
3359:
3356:
3354:
3351:
3349:
3346:
3342:
3338:
3334:
3332:
3329:
3327:
3324:
3323:
3320:
3317:
3315:
3312:
3310:
3307:
3305:
3302:
3298:
3294:
3290:
3288:
3285:
3284:
3282:
3277:
3272:
3269:
3265:
3261:
3258:
3245:
3244:
3232:
3212:
3209:
3206:
3195:
3183:
3161:
3158:
3154:
3133:
3113:
3108:
3105:
3101:
3076:
3056:
3052:
3048:
3045:
3042:
3039:
3028:
3012:
2992:
2972:
2948:
2928:
2908:
2897:
2885:
2861:
2840:
2818:
2797:
2787:
2775:
2751:
2740:
2724:
2702:
2697:
2692:
2689:
2665:
2662:
2659:
2656:
2653:
2650:
2627:
2607:
2604:
2601:
2581:
2569:
2566:
2544:
2536:In fact, when
2519:
2468:A real matrix
2443:
2440:
2437:
2434:
2431:
2428:
2425:
2422:
2400:
2395:
2373:
2370:
2367:
2363:
2359:
2337:
2332:
2310:
2307:
2304:
2300:
2296:
2261:
2248:
2223:
2218:directed graph
2196:
2195:
2184:
2179:
2173:
2170:
2168:
2165:
2164:
2161:
2158:
2156:
2153:
2152:
2150:
2145:
2140:
2137:
2133:
2129:
2126:
2086:, 0 <
2082:
2078:
2070:
2066:
2007:
2004:
1989:
1952:
1931:
1910:
1907:
1874:
1866:
1863:
1861:
1858:
1857:
1854:
1851:
1849:
1846:
1845:
1841:
1837:
1834:
1809:
1801:
1798:
1796:
1793:
1792:
1789:
1786:
1784:
1781:
1780:
1776:
1772:
1769:
1752:
1749:
1736:
1733:
1730:
1725:
1721:
1682:
1681:
1670:
1665:
1662:
1658:
1652:
1648:
1642:
1638:
1634:
1631:
1628:
1623:
1620:
1616:
1610:
1606:
1600:
1596:
1584:
1573:
1568:
1564:
1558:
1554:
1548:
1543:
1540:
1537:
1533:
1528:
1522:
1518:
1512:
1509:
1505:
1499:
1495:
1489:
1484:
1481:
1478:
1475:
1472:
1468:
1462:
1459:
1456:
1453:
1450:
1444:
1441:
1438:
1435:
1429:
1426:
1423:
1420:
1416:
1407:
1404:
1401:
1397:
1393:
1387:
1382:
1378:
1372:
1369:
1364:
1360:
1351:
1348:
1345:
1342:
1339:
1333:
1330:
1327:
1324:
1318:
1315:
1312:
1309:
1305:
1296:
1293:
1290:
1286:
1282:
1279:
1262:
1251:
1246:
1242:
1237:
1231:
1227:
1223:
1220:
1217:
1212:
1208:
1202:
1197:
1194:
1191:
1187:
1181:
1178:
1175:
1171:
1165:
1161:
1157:
1154:
1121:
1110:
1105:
1101:
1095:
1091:
1085:
1080:
1077:
1074:
1070:
1065:
1059:
1055:
1049:
1046:
1042:
1036:
1032:
1026:
1021:
1018:
1015:
1012:
1009:
1005:
999:
996:
993:
989:
983:
980:
977:
973:
969:
963:
958:
954:
948:
945:
940:
936:
927:
924:
921:
917:
911:
908:
905:
901:
897:
891:
886:
882:
876:
873:
868:
864:
855:
852:
849:
845:
839:
836:
833:
829:
825:
822:
793:
771:
762:
747:
730:
718:
709:
688:
650:
646:
642:
639:
634:
630:
625:
619:
615:
609:
606:
603:
599:
588:
578:
548:
519:
507:are positive:
487:
480:
470:
457:associated to
439:
419:
416:
413:
410:
384:absolute value
327:
324:
321:
318:
315:
312:
309:
289:
286:
281:
278:
274:
253:
250:
247:
227:
222:
219:
215:
211:
208:
205:
193:
190:
162:absolute value
104:
101:
15:
9:
6:
4:
3:
2:
10391:
10380:
10377:
10375:
10372:
10370:
10369:Matrix theory
10367:
10366:
10364:
10352:
10348:
10344:
10341:
10337:
10332:
10328:
10323:
10319:
10318:
10313:
10308:
10306:
10302:
10298:
10294:
10291:
10290:0-387-19832-6
10287:
10283:
10282:
10277:
10274:
10270:
10267:
10263:
10260:
10256:
10253:
10249:
10245:
10242:
10239:
10238:0-89871-321-8
10235:
10231:
10227:
10223:
10222:
10213:
10209:
10204:
10199:
10195:
10191:
10187:
10183:
10178:
10175:
10171:
10167:
10163:
10159:
10155:
10151:
10147:
10143:
10138:
10134:
10130:
10125:
10122:
10120:0-471-83966-3
10116:
10112:
10107:
10104:on 2010-03-07
10100:
10096:
10090:
10083:
10082:
10076:
10073:
10069:
10065:
10061:
10057:
10053:
10048:
10045:
10041:
10037:
10031:
10027:
10023:
10019:
10014:
10010:
10004:
10000:
9999:
9994:
9990:
9986:
9982:
9978:
9974:
9970:
9966:
9962:
9961:
9956:
9952:
9949:
9945:
9940:
9935:
9931:
9927:
9923:
9919:
9918:
9913:
9909:
9908:Perron, Oskar
9905:
9904:
9893:
9889:
9883:
9875:
9869:
9851:
9844:
9840:
9836:
9831:
9823:
9817:
9799:
9792:
9788:
9784:
9779:
9772:
9768:
9764:
9760:
9753:
9746:
9740:
9736:
9731:
9723:
9717:
9713:
9706:
9698:
9692:
9688:
9684:
9678:
9670:
9664:
9659:
9658:
9652:
9648:
9642:
9635:
9630:
9622:
9616:
9598:
9591:
9587:
9583:
9578:
9570:
9564:
9546:
9539:
9535:
9531:
9526:
9520:
9516:
9511:
9503:
9497:
9479:
9472:
9468:
9464:
9459:
9451:
9445:
9427:
9420:
9416:
9412:
9407:
9400:
9395:
9387:
9381:
9363:
9356:
9352:
9348:
9343:
9335:
9331:
9327:
9321:
9317:
9316:
9308:
9300:
9298:0-471-83966-3
9294:
9290:
9286:
9280:
9273:
9271:9783540627388
9267:
9263:
9262:
9254:
9248:
9244:
9239:
9231:
9225:
9207:
9200:
9195:
9191:
9186:
9178:
9172:
9154:
9147:
9143:
9139:
9134:
9132:
9130:
9128:
9118:
9113:
9109:
9105:
9101:
9094:
9085:
9076:
9067:
9059:
9055:
9051:
9044:
9036:
9032:
9025:
9019:
9015:
9010:
9002:
8996:
8981:
8975:
8971:
8970:
8965:
8961:
8957:
8952:
8944:
8938:
8920:
8913:
8909:
8905:
8900:
8892:
8886:
8868:
8861:
8857:
8853:
8848:
8840:
8836:
8832:
8828:
8824:
8820:
8813:
8809:
8799:
8796:
8790:
8787:
8784:
8780:
8777:
8774:
8771:
8768:
8765:
8763:
8760:
8757:
8754:
8751:
8748:
8747:
8741:
8739:
8735:
8731:
8727:
8723:
8719:
8715:
8713:
8709:
8704:
8702:
8698:
8694:
8690:
8686:
8682:
8678:
8674:
8670:
8666:
8656:
8654:
8650:
8646:
8641:
8637:
8633:
8616:
8608:
8603:
8598:
8593:
8588:
8581:
8576:
8571:
8566:
8561:
8554:
8549:
8544:
8539:
8534:
8527:
8522:
8517:
8512:
8507:
8500:
8495:
8490:
8485:
8480:
8472:
8463:
8446:
8438:
8433:
8428:
8421:
8416:
8411:
8404:
8399:
8394:
8386:
8377:
8360:
8352:
8347:
8342:
8339:
8329:
8324:
8319:
8312:
8307:
8302:
8294:
8285:
8268:
8260:
8255:
8250:
8243:
8238:
8233:
8226:
8221:
8216:
8208:
8199:
8196:The matrices
8189:
8187:
8183:
8179:
8175:
8171:
8167:
8163:
8160: −
8159:
8155:
8151:
8147:
8143:
8139:
8136: =
8135:
8131:
8112:
8108:
8102:
8099:
8095:
8089:
8084:
8080:
8074:
8071:
8067:
8057:
8053:
8049:
8045:
8041:
8037:
8033:
8029:
8025:
8021:
8017:
8013:
8009:
8005:
8001:
7991:
7989:
7985:
7981:
7977:
7973:
7969:
7965:
7961:
7957:
7947:
7945:
7941:
7937:
7933:
7928:
7926:
7922:
7918:
7914:
7910:
7906:
7902:
7898:
7894:
7890:
7886:
7882:
7878:
7874:
7870:
7867:= 0 if λ ≠ ρ(
7866:
7862:
7858:
7854:
7850:
7846:
7842:
7838:
7834:
7830:
7826:
7822:
7818:
7814:
7810:
7805:
7802:
7800:
7785:
7783:
7779:
7774:
7772:
7768:
7764:
7759:
7754:
7750:
7747:) is 1. Then
7746:
7739:
7735:
7731:
7727:
7723:
7704:
7701:
7697:
7691:
7688:
7684:
7678:
7674:
7668:
7656:
7655:
7654:
7651:
7649:
7627:
7619:
7606:
7581:
7571:
7568:
7564:
7553:
7548:
7545:
7542:
7538:
7532:
7529:
7526:
7523:
7520:
7512:
7499:
7485:
7484:infinity norm
7463:
7455:
7447:
7438:
7429:
7426:
7418:
7410:
7401:
7392:
7389:
7381:
7375:
7350:
7328:
7301:
7293:
7287:
7275:
7270:
7268:
7244:
7241:
7237:
7226:
7222:
7216:
7207:
7198:
7171:
7162:
7143:
7138:
7135:
7131:
7125:
7121:
7115:
7106:
7102:
7095:
7094:
7093:
7091:
7087:
7077:
7075:
7071:
7067:
7064:
7060:
7056:
7052:
7048:
7044:
7040:
7037:takes a form
7036:
7032:
7028:
7024:
7020:
7016:
7012:
7008:
7004:
7000:
6995:
6993:
6989:
6985:
6981:
6977:
6973:
6969:
6965:
6962:
6958:
6954:
6951:
6947:
6943:
6939:
6935:
6931:
6927:
6923:
6919:
6915:
6912:
6908:
6904:
6900:
6895:
6893:
6889:
6885:
6881:
6876:
6872:
6868:
6861:
6857:
6853:
6849:
6842:
6837:
6833:
6826:
6822:
6818:
6813:
6811:
6807:
6803:
6798:
6796:
6792:
6788:
6780:
6776:
6773:
6770:
6766:
6762:
6758:
6755:
6752:
6748:
6744:
6740:
6736:
6733:The image of
6732:
6729:
6725:
6721:
6717:
6713:
6709:
6705:
6702:
6699:
6695:
6691:
6687:
6686:
6685:
6683:
6679:
6672:
6668:
6662:
6660:
6656:
6652:
6648:
6644:
6640:
6636:
6632:
6628:
6624:
6620:
6617:and conclude
6616:
6612:
6608:
6604:
6600:
6597:by the value
6596:
6591:
6589:
6585:
6581:
6577:
6570:
6565:
6561:
6556:
6551:
6547:
6543:
6539:
6529:
6527:
6523:
6519:
6515:
6511:
6508:
6504:
6501:
6497:
6493:
6489:
6485:
6482:
6478:
6475:
6471:
6467:
6463:
6459:
6455:
6451:
6447:
6443:
6439:
6434:
6430:
6428:
6424:
6414:
6412:
6408:
6403:
6401:
6398:
6394:
6390:
6386:
6383:
6380:
6376:
6372:
6368:
6364:
6360:
6353:
6334:
6329:
6321:
6317:
6311:
6302:
6299:
6296:
6292:
6288:
6281:
6277:
6270:
6265:
6260:
6254:
6248:
6243:
6236:
6231:
6225:
6219:
6214:
6210:
6202:
6201:
6200:
6195:
6191:
6184:
6163:
6159:
6150:
6141:
6134:
6128:
6113:
6109:
6087:
6079:
6075:
6066:
6046:
6042:
6035:
6022:
6017:
6013:
6006:
5993:
5982:
5980:
5976:
5972:
5968:
5963:
5961:
5953:
5952:infinity norm
5948:
5946:
5941:
5939:
5935:
5931:
5927:
5923:
5919:
5915:
5911:
5907:
5903:
5899:
5895:
5891:
5887:
5883:
5879:
5875:
5871:
5866:
5864:
5854:
5851:
5847:
5844:>0, hence
5842:
5838:
5834:
5829:
5824:
5819:
5816:
5811:
5807:
5804:
5799:
5794:
5791:, such that (
5790:
5786:
5783:>0. Hence
5781:
5777:
5773:
5768:
5763:
5759:
5756:
5751:
5747:
5744:
5739:
5734:
5730:
5726:
5722:
5717:
5712:
5709:, such that (
5708:
5705:there exists
5704:
5696:
5692:
5688:
5684:
5683:
5682:
5679:
5675:
5673:
5669:
5659:
5652:
5648:
5644:
5639:
5635:
5630:
5626:
5620:
5616:
5612:
5608:
5603:
5601:
5597:
5594:the dominant
5593:
5583:
5580:
5578:
5574:
5570:
5566:
5562:
5560:
5556:
5552:
5548:
5544:
5540:
5530:
5527:
5525:
5521:
5517:
5513:
5509:
5505:
5502:
5498:
5494:
5490:
5486:
5482:
5478:
5474:
5471: −
5470:
5466:
5462:
5458:
5454:
5450:
5447: −
5446:
5442:
5438:
5434:
5431:
5427:
5423:
5419:
5415:
5405:
5403:
5399:
5395:
5391:
5386:
5384:
5380:
5376:
5372:
5368:
5358:
5356:
5352:
5348:
5341:Proof methods
5338:
5336:
5332:
5331:arrow of time
5328:
5324:
5320:
5316:
5312:
5308:
5302:
5292:
5290:
5286:
5285:Markov chains
5276:
5274:
5270:
5266:
5259:
5255:
5251:
5247:
5237:
5235:
5231:
5227:
5222:
5220:
5210:
5208:
5201:
5194:
5189:
5187:
5183:
5179:
5175:
5171:
5167:
5163:
5159:
5155:
5151:
5147:
5140:
5136:
5132:
5128:
5124:
5120:
5113:
5106:
5102:
5097:
5095:
5088:
5084:
5080:
5076:
5069:
5043:
5033:
5029:
5023:
5018:
5013:
5008:
5001:
4996:
4991:
4986:
4981:
4974:
4968:
4963:
4958:
4951:
4946:
4941:
4934:
4930:
4924:
4917:
4912:
4907:
4902:
4895:
4891:
4882:
4873:
4870:
4869:
4868:
4867:
4866:
4865:
4864:
4863:
4862:
4860:
4856:
4846:
4822:
4814:
4809:
4804:
4799:
4794:
4789:
4782:
4777:
4772:
4767:
4762:
4757:
4750:
4744:
4739:
4734:
4729:
4722:
4717:
4712:
4707:
4702:
4697:
4690:
4685:
4680:
4675:
4670:
4665:
4658:
4653:
4648:
4643:
4638:
4633:
4625:
4621:
4618:
4611:
4610:
4602:
4598:
4594:
4590:
4586:
4582:
4578:
4574:
4570:
4566:
4565:
4564:
4562:
4558:
4554:
4550:
4546:
4542:
4538:
4530:
4519:
4516:Moreover, if
4514:
4510:
4505:
4501:
4497:
4493:
4489:
4487:is primitive.
4486:
4482:
4478:
4475:
4471:
4467:
4463:
4460:
4456:
4452:
4448:
4445:
4441:
4425:
4417:
4413:
4409:
4403:
4398:
4394:
4389:
4383:
4379:
4373:
4370:
4367:
4364:
4361:
4358:
4355:
4352:
4349:
4345:
4341:
4337:
4333:
4322:
4309:
4306:
4303:
4300:
4285:
4281:
4277:
4252:
4248:
4244:
4241:
4238:is period of
4237:
4233:
4229:
4225:
4221:
4200:
4192:
4188:
4182:
4177:
4172:
4167:
4160:
4154:
4149:
4144:
4137:
4132:
4127:
4120:
4116:
4110:
4103:
4098:
4093:
4088:
4081:
4077:
4070:
4065:
4060:
4057:
4053:
4047:
4043:
4039:
4031:
4027:
4023:
4020:
4016:
4012:
4008:
4005:
4001:
3997:
3993:
3989:
3985:
3981:
3977:
3973:
3969:
3965:
3962:
3958:
3954:
3950:
3949:
3948:
3946:
3936:
3934:
3930:
3925:
3921:
3904:
3896:
3891:
3886:
3879:
3874:
3869:
3862:
3857:
3852:
3844:
3840:
3837:
3812:
3807:
3804:
3800:
3794:
3790:
3784:
3776:
3773:
3770:
3765:
3762:
3758:
3752:
3748:
3742:
3730:
3729:
3728:
3727:
3723:
3722:
3718:
3702:
3682:
3679:
3674:
3670:
3649:
3641:
3626:
3622:
3617:
3611:
3598:
3586:
3562:
3553:
3529:
3527:
3523:
3522:
3521:
3507:
3494:
3480:
3475:
3469:
3464:
3459:
3454:
3449:
3442:
3438:
3428:
3425:
3422:
3418:
3412:
3407:
3402:
3397:
3392:
3385:
3379:
3374:
3369:
3364:
3357:
3352:
3347:
3340:
3336:
3330:
3325:
3318:
3313:
3308:
3303:
3296:
3292:
3286:
3280:
3275:
3270:
3267:
3263:
3259:
3256:
3249:
3248:
3247:
3246:
3230:
3210:
3207:
3204:
3196:
3181:
3159:
3156:
3152:
3131:
3111:
3106:
3103:
3099:
3090:
3074:
3054:
3050:
3046:
3043:
3040:
3037:
3029:
3026:
3025:root of unity
3010:
2990:
2970:
2962:
2946:
2926:
2906:
2898:
2883:
2875:
2859:
2795:
2788:
2773:
2765:
2749:
2741:
2738:
2722:
2700:
2690:
2687:
2679:
2678:
2677:
2663:
2660:
2654:
2648:
2641:
2625:
2605:
2602:
2599:
2579:
2565:
2562:
2558:
2556:
2552:
2547:
2543:
2539:
2535:
2533:
2527:
2524:> 0. When
2522:
2517:
2513:
2509:
2505:
2504:
2498:
2494:
2489:
2487:
2483:
2479:
2475:
2471:
2466:
2464:
2459:
2457:
2438:
2435:
2429:
2426:
2420:
2398:
2368:
2365:
2335:
2305:
2302:
2286:
2282:
2281:Definition 4:
2278:
2276:
2271:
2269:
2264:
2260:
2256:
2251:
2247:
2243:
2239:
2235:
2231:
2226:
2222:
2219:
2215:
2211:
2210:Definition 3:
2207:
2205:
2201:
2182:
2177:
2171:
2166:
2159:
2154:
2148:
2143:
2138:
2135:
2131:
2127:
2124:
2117:
2116:
2115:
2113:
2110:
2106:
2103:
2102:Definition 2:
2099:
2097:
2093:
2089:
2081:
2077:
2069:
2065:
2061:
2057:
2056:proper subset
2053:
2049:
2045:
2042:
2038:
2036:
2032:
2029:. The matrix
2028:
2025:
2021:
2018: ×
2017:
2013:
2003:
2000:
1987:
1978:
1974:
1970:
1966:
1950:
1942:
1929:
1908:
1905:
1896:
1891:
1889:
1872:
1864:
1859:
1852:
1847:
1839:
1835:
1832:
1824:
1807:
1799:
1794:
1787:
1782:
1774:
1770:
1767:
1759:
1748:
1734:
1731:
1728:
1723:
1719:
1710:
1706:
1702:
1698:
1693:
1691:
1687:
1668:
1663:
1660:
1656:
1650:
1646:
1640:
1632:
1629:
1626:
1621:
1618:
1614:
1608:
1604:
1598:
1585:
1571:
1566:
1562:
1556:
1552:
1546:
1541:
1538:
1535:
1531:
1526:
1520:
1516:
1510:
1507:
1503:
1497:
1493:
1487:
1482:
1479:
1476:
1473:
1470:
1466:
1460:
1457:
1454:
1451:
1448:
1442:
1439:
1436:
1433:
1427:
1424:
1421:
1418:
1405:
1402:
1399:
1391:
1385:
1376:
1370:
1367:
1358:
1349:
1346:
1343:
1340:
1337:
1331:
1328:
1325:
1322:
1316:
1313:
1310:
1307:
1294:
1291:
1288:
1280:
1277:
1269:
1267:
1263:
1249:
1244:
1240:
1235:
1229:
1221:
1218:
1210:
1206:
1200:
1195:
1192:
1189:
1185:
1179:
1176:
1173:
1163:
1155:
1152:
1144:
1140:
1136:
1134:
1130:
1126:
1122:
1108:
1103:
1099:
1093:
1089:
1083:
1078:
1075:
1072:
1068:
1063:
1057:
1053:
1047:
1044:
1040:
1034:
1030:
1024:
1019:
1016:
1013:
1010:
1007:
1003:
997:
994:
991:
981:
978:
975:
967:
961:
952:
946:
943:
934:
925:
922:
919:
909:
906:
903:
895:
889:
880:
874:
871:
862:
853:
850:
847:
837:
834:
831:
823:
820:
812:
808:
804:
802:
798:
794:
791:
787:
783:
779:
774:
770:
765:
760:
756:
752:
748:
745:
741:
737:
733:
726:
721:
717:
712:
707:
703:
699:
695:
693:
689:
686:
682:
678:
674:
670:
666:
648:
644:
640:
637:
632:
628:
623:
617:
613:
601:
589:
587:
583:
579:
576:
572:
568:
564:
560:
556:
555:Perron vector
551:
547:
543:
539:
535:
531:
527:
522:
518:
514:
510:
506:
502:
498:
494:
490:
486:
479:
475:
471:
468:
464:
460:
456:
452:
448:
444:
440:
437:
436:Perron number
433:
414:
408:
401:
397:
393:
389:
385:
381:
377:
373:
369:
366:), such that
365:
361:
357:
353:
349:
346:, called the
345:
341:
340:
339:
325:
322:
319:
316:
313:
310:
307:
287:
284:
279:
276:
272:
251:
248:
245:
220:
217:
213:
206:
203:
189:
187:
183:
179:
175:
171:
167:
163:
159:
155:
151:
147:
143:
139:
135:
132:
131:square matrix
128:
124:
120:
116:
112:
111:
100:
98:
97:Edmund Landau
94:
90:
86:
82:
78:
74:
70:
66:
65:Markov chains
62:
58:
54:
50:
46:
42:
38:
34:
30:
26:
22:
21:matrix theory
10350:
10339:
10335:
10330:
10326:
10315:
10296:
10280:
10272:
10265:
10258:
10251:
10248:Gordon Royle
10244:Chris Godsil
10229:
10185:
10181:
10157:
10153:
10132:
10128:
10110:
10099:the original
10080:
10058:(1): 80–93,
10055:
10051:
10017:
9997:
9984:
9972:
9958:
9921:
9915:
9882:
9857:. Retrieved
9850:the original
9830:
9805:. Retrieved
9798:the original
9778:
9765:(1): 29–35,
9762:
9758:
9745:
9730:
9711:
9705:
9686:
9677:
9656:
9641:
9629:
9604:. Retrieved
9597:the original
9577:
9552:. Retrieved
9545:the original
9525:
9510:
9485:. Retrieved
9478:the original
9458:
9433:. Retrieved
9426:the original
9406:
9394:
9369:. Retrieved
9362:the original
9342:
9318:. Elsevier.
9314:
9307:
9288:
9285:Minc, Henryk
9279:
9264:, Springer,
9260:
9253:
9238:
9213:. Retrieved
9206:the original
9185:
9160:. Retrieved
9153:the original
9107:
9103:
9093:
9084:
9075:
9066:
9057:
9053:
9043:
9034:
9030:
9024:
9009:
8984:. Retrieved
8968:
8951:
8926:. Retrieved
8919:the original
8908:8.3.7 p. 683
8899:
8874:. Retrieved
8867:the original
8856:8.3.6 p. 681
8847:
8822:
8818:
8812:
8737:
8733:
8729:
8725:
8721:
8717:
8716:
8711:
8705:
8700:
8696:
8692:
8688:
8685:reducibility
8684:
8680:
8677:non-negative
8676:
8672:
8668:
8664:
8662:
8652:
8648:
8644:
8639:
8635:
8631:
8461:
8375:
8283:
8197:
8195:
8185:
8181:
8177:
8173:
8169:
8165:
8161:
8157:
8153:
8149:
8145:
8141:
8137:
8133:
8132:is given by
8129:
8055:
8051:
8047:
8043:
8039:
8035:
8031:
8027:
8023:
8019:
8015:
8011:
8007:
8003:
7999:
7997:
7987:
7983:
7979:
7975:
7971:
7967:
7963:
7959:
7955:
7953:
7943:
7939:
7935:
7931:
7929:
7924:
7920:
7916:
7912:
7908:
7904:
7900:
7896:
7892:
7888:
7884:
7880:
7876:
7872:
7868:
7864:
7863:which means
7860:
7856:
7852:
7848:
7844:
7840:
7836:
7832:
7828:
7824:
7820:
7816:
7812:
7808:
7806:
7803:
7796:
7781:
7775:
7770:
7766:
7762:
7757:
7752:
7748:
7741:
7737:
7733:
7729:
7725:
7721:
7719:
7652:
7647:
7342:because, if
7271:
7160:
7158:
7089:
7085:
7083:
7073:
7069:
7065:
7062:
7058:
7054:
7050:
7046:
7042:
7038:
7034:
7030:
7026:
7022:
7018:
7014:
7010:
7006:
7002:
6998:
6997:Denoting by
6996:
6991:
6987:
6986:lies in the
6983:
6979:
6975:
6971:
6967:
6963:
6960:
6956:
6952:
6949:
6945:
6941:
6937:
6933:
6929:
6925:
6921:
6917:
6913:
6910:
6906:
6902:
6898:
6896:
6891:
6887:
6883:
6879:
6874:
6870:
6866:
6859:
6855:
6851:
6847:
6840:
6835:
6831:
6824:
6820:
6816:
6814:
6805:
6801:
6799:
6794:
6786:
6784:
6778:
6771:
6768:
6764:
6760:
6756:
6750:
6746:
6742:
6738:
6734:
6727:
6723:
6719:
6715:
6711:
6703:
6697:
6693:
6689:
6681:
6677:
6675:
6670:
6666:
6658:
6655:f(x) = ξ ≤ r
6654:
6650:
6646:
6642:
6638:
6634:
6630:
6626:
6622:
6618:
6614:
6613:, we obtain
6610:
6606:
6602:
6598:
6594:
6592:
6587:
6579:
6572:
6568:
6563:
6559:
6554:
6549:
6545:
6541:
6537:
6535:
6525:
6521:
6517:
6513:
6509:
6506:
6502:
6499:
6495:
6491:
6487:
6483:
6480:
6476:
6473:
6469:
6465:
6461:
6457:
6453:
6449:
6445:
6441:
6437:
6435:
6431:
6426:
6422:
6420:
6410:
6406:
6404:
6399:
6396:
6392:
6388:
6384:
6381:
6378:
6374:
6370:
6366:
6362:
6358:
6351:
6349:
6193:
6189:
6182:
5983:
5978:
5974:
5970:
5966:
5964:
5959:
5949:
5944:
5942:
5937:
5933:
5925:
5921:
5917:
5913:
5909:
5905:
5901:
5897:
5893:
5889:
5885:
5881:
5877:
5873:
5869:
5867:
5862:
5860:
5849:
5845:
5840:
5836:
5832:
5827:
5822:
5817:
5814:
5809:
5805:
5802:
5797:
5792:
5788:
5784:
5779:
5775:
5771:
5766:
5761:
5757:
5754:
5749:
5745:
5742:
5737:
5732:
5728:
5724:
5720:
5715:
5710:
5706:
5702:
5700:
5694:
5690:
5686:
5680:
5676:
5671:
5664:
5657:
5650:
5637:
5633:
5628:
5624:
5618:
5614:
5610:
5607:power method
5604:
5599:
5591:
5589:
5581:
5576:
5572:
5568:
5564:
5563:
5558:
5554:
5550:
5546:
5545:, such that
5542:
5538:
5536:
5528:
5523:
5519:
5515:
5511:
5507:
5503:
5496:
5492:
5488:
5484:
5480:
5476:
5472:
5468:
5464:
5460:
5456:
5452:
5448:
5444:
5440:
5436:
5433:
5429:
5425:
5421:
5417:
5413:
5411:
5401:
5397:
5393:
5389:
5387:
5382:
5374:
5370:
5366:
5364:
5344:
5319:David Ruelle
5314:
5304:
5282:
5261:
5257:
5253:
5249:
5243:
5233:
5229:
5225:
5223:
5216:
5203:
5196:
5192:
5190:
5185:
5181:
5177:
5173:
5169:
5165:
5161:
5157:
5153:
5149:
5142:
5138:
5134:
5130:
5126:
5122:
5115:
5108:
5104:
5100:
5098:
5090:
5086:
5082:
5078:
5071:
5067:
5065:
4871:
4858:
4854:
4852:
4843:
4840:Applications
4600:
4596:
4592:
4588:
4584:
4580:
4576:
4572:
4568:
4560:
4556:
4552:
4548:
4547:, and hence
4544:
4540:
4536:
4534:
4521:
4517:
4512:
4508:
4503:
4499:
4495:
4491:
4484:
4480:
4473:
4469:
4465:
4458:
4450:
4446:
4443:
4439:
4287:
4283:
4279:
4254:
4250:
4246:
4239:
4235:
4231:
4227:
4223:
4216:
4029:
4025:
4018:
4014:
4010:
4006:
4003:
3999:
3995:
3991:
3987:
3983:
3979:
3975:
3971:
3967:
3960:
3952:
3944:
3942:
3932:
3928:
3923:
3919:
3830:The example
3829:
3588:
3555:
3531:
3524:
3499:
2960:
2919:has exactly
2873:
2736:
2571:
2563:
2559:
2554:
2550:
2545:
2541:
2537:
2531:
2529:
2525:
2520:
2515:
2511:
2502:
2500:
2496:
2492:
2490:
2485:
2481:
2477:
2473:
2469:
2467:
2462:
2461:A matrix is
2460:
2280:
2279:
2274:
2272:
2262:
2258:
2254:
2249:
2245:
2241:
2237:
2233:
2229:
2224:
2220:
2213:
2209:
2208:
2203:
2199:
2197:
2111:
2104:
2101:
2100:
2095:
2091:
2087:
2079:
2075:
2067:
2063:
2059:
2047:
2043:
2040:
2039:
2034:
2030:
2026:
2019:
2015:
2011:
2009:
1980:
1972:
1964:
1922:
1894:
1892:
1887:
1822:
1757:
1754:
1708:
1704:
1700:
1696:
1694:
1685:
1683:
1264:
1142:
1138:
1123:
810:
806:
795:
789:
781:
777:
772:
768:
763:
758:
754:
750:
743:
735:
728:
724:
719:
715:
710:
705:
701:
697:
690:
684:
680:
672:
668:
664:
585:
581:
574:
570:
566:
562:
558:
554:
549:
545:
541:
537:
533:
529:
525:
520:
516:
512:
508:
504:
500:
496:
493:
488:
484:
477:
473:
466:
462:
458:
450:
442:
431:
430:is equal to
398:. Thus, the
395:
391:
387:
375:
371:
367:
363:
359:
355:
351:
347:
343:
195:
169:
157:
153:
149:
133:
115:non-negative
114:
109:
106:
27:, proved by
24:
18:
10295:Seneta, E.
10257:A. Graham,
10135:: 213–219,
10052:SIAM Review
9837:, pp.
9785:, pp.
9584:, pp.
9532:, pp.
9465:, pp.
9413:, pp.
9349:, pp.
9192:, pp.
9140:, pp.
9014:Keener 1993
8960:15.2 p. 167
8906:, pp.
8854:, pp.
8689:irreducible
8659:Terminology
7041:, for some
6815:Given that
6635:0 ≤ ξx ≤ Ax
5731:such that (
5643:eigenvector
5596:eigenvector
4269:x + ... + c
2899:The matrix
2680:The number
2514:such that (
2035:irreducible
1975:called the
1895:irreducible
776:taken over
348:Perron root
127:eigenvalues
53:eigenvector
10363:Categories
10334:has order
9901:References
9859:2010-03-07
9835:Meyer 2000
9807:2010-03-07
9783:Meyer 2000
9737:, p.
9634:Varga 2002
9606:2010-03-07
9582:Meyer 2000
9554:2010-03-07
9530:Meyer 2000
9517:, p.
9487:2010-03-07
9463:Meyer 2000
9435:2010-03-07
9411:Meyer 2000
9371:2010-03-07
9347:Meyer 2000
9245:, p.
9215:2010-03-07
9190:Meyer 2000
9162:2010-03-07
9138:Meyer 2000
9016:, p.
8986:2016-10-31
8958:, p.
8928:2010-03-07
8904:Meyer 2000
8876:2010-03-07
8852:Meyer 2000
8002:) = 1 and
7728:such that
6869:), ... , (
6810:polynomial
6578:≠ 0. Then
6571:such that
6548:such that
5973:. Letting
5912:such that
5647:eigenvalue
5428:such that
5114:so if any
4017:equals to
3662:such that
2530:period of
2506:to be the
2240:to vertex
2216:a certain
2048:coordinate
1969:roots of 1
734:≠ 0. Then
727:such that
455:eigenspace
378:(possibly
129:of a real
61:ergodicity
49:eigenvalue
10322:EMS Press
10212:122189604
10174:120958677
9995:(2000) ,
9987:: 514–518
9975:: 471–476
9963:: 456–477
9948:123460172
9334:922964628
9110:: 81–94.
9060:: 192–202
9037:: 366–369
8995:cite book
8839:0309-166X
8701:connected
8340:−
8100:−
8096:λ
8081:∑
8072:−
7994:Cyclicity
7698:≤
7675:∑
7628:λ
7620:≥
7615:∞
7611:‖
7604:‖
7539:∑
7530:≤
7524:≤
7508:∞
7464:λ
7427:λ
7382:≥
7379:‖
7373:‖
7329:λ
7302:λ
7294:≥
7291:‖
7285:‖
7223:∑
7208:≤
7199:λ
7172:λ
7122:∑
7107:≤
6460:), where
6318:λ
6300:−
6293:λ
6278:λ
6249:λ
6232:λ
6138:‖
6132:‖
6129:≤
6124:∞
6120:‖
6106:‖
6097:⇒
6057:∞
6053:‖
6039:‖
6036:≥
6031:∞
6027:‖
6010:‖
6002:∞
5998:‖
5991:‖
5024:⋯
5002:∗
4997:⋯
4975:⋮
4969:⋮
4964:⋮
4959:⋮
4952:∗
4947:⋯
4942:∗
4918:∗
4913:⋯
4908:∗
4903:∗
4810:⋯
4778:⋯
4751:⋮
4745:⋮
4740:⋮
4735:⋮
4730:⋮
4718:⋯
4686:⋯
4654:⋯
4535:A matrix
4346:∑
4329:∞
4326:→
4296:, ... , k
4183:…
4161:⋮
4155:⋮
4150:⋮
4145:⋮
4133:…
4099:…
4058:−
3791:∑
3777:≤
3771:≤
3749:∑
3680:≠
3465:…
3426:−
3413:…
3386:⋮
3380:⋮
3375:⋮
3370:⋮
3365:⋮
3353:…
3314:…
3268:−
3243:such that
3182:ω
3160:ω
3107:ω
3047:π
3038:ω
2691:∈
2649:ρ
2603:×
2555:aperiodic
2474:primitive
2463:reducible
2430:
2424:↦
2413:given by
2228:. It has
2144:≠
2136:−
1951:ω
1906:ω
1690:chapter 8
1647:∑
1633:≤
1627:≤
1605:∑
1532:∑
1467:∑
1452:∘
1381:⊤
1363:⊤
1341:∘
1186:∑
1133:Friedland
1069:∑
1004:∑
957:⊤
939:⊤
885:⊤
867:⊤
608:∞
605:→
409:ρ
323:≤
311:≤
249:×
103:Statement
10349:(2002),
10087:, SIAM,
9910:(1907),
9868:cite web
9816:cite web
9653:(1992).
9615:cite web
9563:cite web
9496:cite web
9444:cite web
9415:page 679
9380:cite web
9287:(1988).
9224:cite web
9171:cite web
8937:cite web
8885:cite web
8767:P-matrix
8762:M-matrix
8744:See also
8673:positive
8669:positive
8046:, ....,
7895:so that
7503:‖
7497:‖
6916: ;
6830:, ... ,
6763:, where
6653:. Hence
6625:and let
6615:f(v) = r
6472:), then
6365:| (for |
5981:, then:
5969:= 1, or
5914:u=v- α w
5439:and set
5396:, where
5256:and arc
5176:+ ... +
4308:averages
4234:, where
4215:, where
3998:), then
3003:with an
1921:, where
1758:or equal
1129:Varadhan
797:Birkhoff
536: :
142:spectrum
123:positive
119:matrices
110:positive
93:PageRank
10072:2132526
10044:7646929
8736:or the
7778:Collatz
7047:(a w v)
7039:(a v w)
6649:, then
6584:maximum
6505:, also
6387:, then
5377:is the
5351:Collatz
5317:(after
5168:is 1 +
4575:, then
4498:, then
4490:If 0 ≤
4453:is the
4255:= x + c
3992:μ=ρ(A)e
3974:, then
3717:maximum
3695:. Then
3526:Collatz
3089:similar
2959:is the
2939:(where
2074:, ...,
1268:formula
1266:Fiedler
1135:formula
1125:Donsker
803:formula
786:minimum
780:. Then
740:maximum
692:Collatz
675:is the
394:| <
380:complex
350:or the
184: (
176: (
166:modulus
39: (
31: (
10303:
10288:
10236:
10210:
10172:
10117:
10091:
10070:
10042:
10032:
10005:
9946:
9718:
9693:
9665:
9332:
9322:
9295:
9268:
8976:
8837:
7875:). If
7482:. The
6955:= lim
6920:= lim
6785:Hence
6777:Hence
6633:gives
6627:ξ=f(x)
6444:) for
6350:hence
6103:
6100:
6094:
6091:
5965:Given
5960:k → ∞
5499:so by
5422:A/ρ(A)
5066:where
4526:< r
4305:Cesàro
3500:where
2961:period
2764:simple
2198:where
2014:be a
1977:period
1446:
1431:
1411:
1335:
1320:
1300:
1137:: Let
805:: Let
753:, let
700:, let
238:be an
35:) and
23:, the
10208:S2CID
10170:S2CID
10102:(PDF)
10085:(PDF)
10068:JSTOR
10040:S2CID
9944:S2CID
9853:(PDF)
9846:(PDF)
9801:(PDF)
9794:(PDF)
9755:(PDF)
9600:(PDF)
9593:(PDF)
9548:(PDF)
9541:(PDF)
9481:(PDF)
9474:(PDF)
9429:(PDF)
9422:(PDF)
9365:(PDF)
9358:(PDF)
9209:(PDF)
9202:(PDF)
9156:(PDF)
9149:(PDF)
9018:p. 80
8922:(PDF)
8915:(PDF)
8870:(PDF)
8863:(PDF)
8805:Notes
7911:. As
7847:then
7740:(say
7033:. So
6884:k → ∞
6789:is a
6706:is a
6694:k → ∞
6659:R ≤ r
6619:r ≤ R
6609:into
6603:R = r
5954:(A/r)
5533:Lemma
5467:thus
5459:then
5325:of a
5271:of a
5180:) so
4591:<
4553:k ≥ m
4494:<
4262:x + c
3970:|<
2024:field
801:Varga
483:,...,
382:) in
10301:ISBN
10286:ISBN
10246:and
10234:ISBN
10115:ISBN
10089:ISBN
10030:ISBN
10003:ISBN
9890:and
9874:link
9822:link
9716:ISBN
9691:ISBN
9663:ISBN
9621:link
9569:link
9502:link
9450:link
9386:link
9330:OCLC
9320:ISBN
9293:ISBN
9266:ISBN
9230:link
9177:link
9001:link
8974:ISBN
8943:link
8891:link
8835:ISSN
8728:and
8720:and
8683:and
8667:and
8152:and
8006:has
7934:and
7855:= ρ(
7823:= ρ(
7272:Any
7013:for
7009:(by
6974:) =
6944:lim
6940:) =
6774:= 1.
6749:for
6692:for
6676:Let
6641:for
6550:f(x)
6498:) =
5979:Av=v
5892:and
5884:and
5689:and
5632:/ |
5518:) ≤
5510:) ≤
5412:Let
5184:and
5152:and
4607:= 0.
4464:Let
4230:and
4007:D AD
3943:Let
3554:let
3208:>
3030:Let
2874:only
2830:and
2638:and
2572:Let
2491:Let
2283:The
2202:and
2010:Let
1699:and
1437:>
1422:>
1403:>
1326:>
1311:>
1292:>
1177:>
995:>
979:>
923:>
907:>
851:>
835:>
809:and
300:for
285:>
196:Let
186:1912
178:1907
136:are
113:and
107:Let
41:1912
33:1907
10198:hdl
10190:doi
10162:doi
10137:doi
10060:doi
10022:doi
9934:hdl
9926:doi
9767:doi
9763:102
9112:doi
8827:doi
8184:as
8156:is
7923:as
7851:= λ
7849:PAx
7843:= λ
7765:of
7751:= (
7665:min
7517:max
7213:max
7112:max
6905:=
6765:v,w
6690:A/r
6479:= (
6395:/ (
6357:= |
6181:So
6147:min
6063:min
5971:A/r
5898:A-r
5826:≥ (
5813:= (
5765:≥ (
5703:i,j
5381:of
5365:If
5291:).
5224:If
5182:PAP
5158:PAP
5139:PAP
5135:PAP
5125:or
5123:PAP
5105:PAP
5083:PAP
4872:PAP
4605:1,1
4567:If
4479:If
4451:v w
4319:lim
4292:, k
4245:If
3951:(I+
3781:max
3739:min
3197:If
3091:to
3087:is
3023:th
2762:is
2553:is
2472:is
2427:exp
2384:on
2350:or
2321:on
2287:of
2273:If
2266:is
2033:is
1967:th
1637:max
1595:min
1415:inf
1396:sup
1304:inf
1285:sup
1170:inf
1160:sup
988:sup
972:inf
916:sup
900:inf
844:inf
828:sup
598:lim
573:or
544:r,
538:w A
513:r v
509:A v
495:of
476:= (
449:of
390:, |
362:or
152:as
63:of
19:In
10365::
10320:,
10314:,
10250:,
10228:,
10206:,
10196:,
10186:52
10184:,
10168:,
10158:48
10156:,
10133:61
10131:,
10066:,
10056:35
10054:,
10038:,
10028:,
9942:,
9932:,
9922:64
9920:,
9914:,
9870:}}
9866:{{
9818:}}
9814:{{
9761:,
9757:,
9649:;
9617:}}
9613:{{
9565:}}
9561:{{
9498:}}
9494:{{
9446:}}
9442:{{
9382:}}
9378:{{
9328:.
9226:}}
9222:{{
9196:.
9173:}}
9169:{{
9126:^
9108:71
9106:.
9102:.
9058:63
9056:,
9052:,
9035:XI
9033:,
8997:}}
8993:{{
8939:}}
8935:{{
8887:}}
8883:{{
8833:.
8821:.
8740:.
8714:.
8687::
8464:=
8460:,
8378:=
8374:,
8286:=
8282:,
8200:=
8042:,
8038:,
8030:=
8024:PA
8022:=
8020:AP
8018:=
7944:vw
7940:wv
7899:=
7865:Px
7861:Px
7853:Px
7841:Ax
7821:PA
7819:=
7817:AP
7753:Aw
7734:rw
7732:=
7730:Aw
7650:.
7076:.
7074:vw
7072:=
7001:,
6980:Pu
6972:Pu
6957:rM
6938:Pu
6928:=
6899:MM
6846:=
6761:vw
6759:=
6753:).
6728:PA
6726:=
6724:AP
6722::
6714:=
6710::
6661:.
6590:.
6558:/
6516:=
6500:λx
6496:Ay
6490:=
6486:)
6474:rx
6456:,
6440:,
6429:.
6391:≥
6377:=
6361:+
5833:ij
5798:ij
5772:ii
5753:=
5738:ii
5716:ij
5634:Ab
5625:Ab
5623:=
5621:+1
5569:AA
5567:=
5557:,
5553:,
5465:λx
5463:=
5461:Ax
5457:λx
5455:=
5453:Ax
5449:εI
5443:=
5404:.
5385:.
5337:.
5264:ij
5258:ij
5172:+
5096:.
4874:=
4861:)
4563::
4513:B.
4507:≤
4310::
4253:)
4002:=
3982:)≤
3935:.
3194:).
2534:.
2521:ii
2270:.
2250:ij
2114::
1965:h'
1747:.
1270::
767:/
714:/
673:vw
669:wv
569:,
565:,
561:,
557:,
540:=
528:≤
515:,
511:=
469:.)
358:,
99:.
79:,
10355:.
10340:h
10338:/
10336:n
10331:j
10327:A
10240:.
10200::
10192::
10164::
10139::
10062::
10024::
9936::
9928::
9894:.
9876:)
9862:.
9824:)
9810:.
9769::
9724:.
9699:.
9671:.
9623:)
9609:.
9571:)
9557:.
9504:)
9490:.
9452:)
9438:.
9388:)
9374:.
9336:.
9301:.
9232:)
9218:.
9179:)
9165:.
9120:.
9114::
9003:)
8989:.
8945:)
8931:.
8893:)
8879:.
8841:.
8829::
8823:5
8785:)
8781:(
8697:A
8693:A
8653:M
8649:M
8645:M
8640:T
8636:P
8632:L
8617:)
8609:0
8604:0
8599:1
8594:0
8589:0
8582:1
8577:0
8572:0
8567:0
8562:0
8555:0
8550:1
8545:0
8540:0
8535:0
8528:0
8523:0
8518:0
8513:0
8508:1
8501:0
8496:0
8491:0
8486:1
8481:0
8473:(
8462:M
8447:)
8439:0
8434:1
8429:1
8422:1
8417:0
8412:1
8405:1
8400:1
8395:0
8387:(
8376:T
8361:)
8353:1
8348:1
8343:1
8330:0
8325:0
8320:1
8313:0
8308:0
8303:1
8295:(
8284:P
8269:)
8261:1
8256:1
8251:1
8244:0
8239:0
8234:1
8227:0
8222:0
8217:1
8209:(
8198:L
8186:n
8182:A
8178:P
8174:A
8170:A
8168:)
8166:P
8162:R
8158:A
8154:R
8150:A
8146:A
8144:)
8142:P
8138:R
8134:A
8130:A
8113:k
8109:R
8103:k
8090:h
8085:1
8075:1
8068:h
8056:A
8052:A
8048:R
8044:R
8040:R
8036:P
8032:P
8028:R
8016:R
8012:P
8008:h
8004:A
8000:A
7988:A
7986:(
7984:ρ
7980:A
7972:A
7968:A
7966:)
7964:P
7960:A
7956:A
7942:/
7936:w
7932:v
7925:n
7921:A
7917:P
7913:n
7909:A
7907:)
7905:P
7901:P
7897:A
7893:A
7891:)
7889:P
7885:P
7881:A
7877:A
7873:A
7869:A
7859:)
7857:A
7845:x
7837:A
7833:P
7829:P
7827:)
7825:A
7813:P
7809:A
7782:x
7771:r
7767:A
7763:i
7758:i
7755:)
7749:r
7744:i
7742:w
7738:w
7726:w
7722:A
7705:.
7702:r
7692:j
7689:i
7685:a
7679:j
7669:i
7648:A
7632:|
7624:|
7607:A
7582:.
7578:|
7572:j
7569:i
7565:a
7560:|
7554:n
7549:1
7546:=
7543:j
7533:m
7527:i
7521:1
7513:=
7500:A
7468:|
7460:|
7456:=
7452:|
7448:x
7444:|
7439:/
7434:|
7430:x
7423:|
7419:=
7415:|
7411:x
7407:|
7402:/
7397:|
7393:x
7390:A
7386:|
7376:A
7351:x
7306:|
7298:|
7288:A
7251:|
7245:j
7242:i
7238:a
7233:|
7227:j
7217:i
7203:|
7195:|
7161:A
7144:.
7139:j
7136:i
7132:a
7126:j
7116:i
7103:r
7090:r
7086:A
7070:P
7066:v
7063:w
7059:M
7055:r
7051:P
7043:a
7035:P
7031:w
7027:P
7023:v
7019:P
7015:M
7011:w
7007:M
7003:r
6999:v
6992:M
6988:r
6984:P
6978:(
6976:r
6970:(
6968:M
6964:u
6961:r
6959:/
6953:u
6950:r
6948:/
6946:M
6942:M
6936:(
6934:M
6930:P
6926:r
6924:/
6922:M
6918:P
6914:M
6911:r
6909:/
6907:M
6903:r
6901:/
6892:M
6888:M
6880:r
6878:/
6875:n
6871:r
6867:r
6865:/
6863:2
6860:r
6856:r
6854:/
6852:M
6848:r
6844:1
6841:r
6836:n
6832:r
6828:1
6825:r
6821:M
6817:M
6806:r
6802:M
6795:r
6787:P
6779:P
6772:v
6769:w
6757:P
6751:A
6747:w
6743:P
6739:v
6735:P
6730:.
6720:A
6716:P
6712:P
6704:P
6700:.
6698:P
6682:r
6678:A
6671:r
6669:/
6667:A
6647:r
6643:A
6639:w
6631:f
6623:x
6611:f
6607:v
6599:R
6595:f
6588:r
6580:f
6575:i
6573:x
6569:i
6564:i
6560:x
6555:i
6546:x
6542:f
6538:A
6526:y
6522:r
6518:λ
6514:r
6510:y
6507:x
6503:y
6494:(
6492:x
6488:y
6484:A
6481:x
6477:y
6470:A
6466:A
6462:x
6458:x
6454:r
6450:y
6446:A
6442:y
6438:λ
6427:A
6423:A
6411:r
6407:r
6400:C
6397:C
6393:J
6389:A
6385:C
6382:A
6379:C
6375:J
6371:k
6367:λ
6363:λ
6359:k
6355:∞
6352:J
6335:,
6330:)
6322:k
6312:0
6303:1
6297:k
6289:k
6282:k
6271:(
6266:=
6261:k
6255:)
6244:0
6237:1
6226:(
6220:=
6215:k
6211:J
6197:∞
6194:A
6190:k
6186:∞
6183:A
6169:)
6164:i
6160:v
6156:(
6151:i
6142:/
6135:v
6114:k
6110:A
6088:,
6085:)
6080:i
6076:v
6072:(
6067:i
6047:k
6043:A
6023:v
6018:k
6014:A
6007:=
5994:v
5975:v
5967:r
5956:∞
5945:r
5938:w
5934:u
5926:u
5922:α
5918:u
5910:α
5906:w
5902:w
5894:r
5890:A
5886:w
5882:v
5878:w
5874:r
5870:v
5863:r
5850:j
5846:v
5841:i
5837:v
5830:)
5828:A
5823:j
5820:)
5818:v
5815:A
5810:j
5806:v
5803:r
5795:)
5793:A
5789:m
5785:r
5780:i
5776:v
5769:)
5767:A
5762:i
5758:v
5755:A
5750:i
5746:v
5743:r
5735:)
5733:A
5729:n
5725:j
5721:v
5713:)
5711:A
5707:m
5695:A
5691:v
5687:A
5672:A
5667:k
5665:b
5661:0
5658:b
5654:0
5651:b
5638:k
5629:k
5619:k
5615:b
5611:A
5600:A
5592:A
5577:A
5573:A
5565:A
5559:A
5555:A
5551:A
5547:A
5543:m
5539:A
5524:A
5522:(
5520:ρ
5516:A
5514:(
5512:ρ
5508:T
5506:(
5504:ρ
5497:A
5493:T
5489:T
5487:(
5485:ρ
5481:m
5477:T
5473:ε
5469:λ
5445:A
5441:T
5437:A
5430:A
5426:m
5418:A
5414:A
5402:A
5398:h
5394:r
5390:h
5383:A
5375:r
5371:r
5367:A
5262:A
5254:n
5250:n
5234:A
5230:A
5226:A
5206:i
5204:B
5199:i
5197:B
5193:A
5186:A
5178:N
5174:N
5170:N
5166:N
5162:N
5156:(
5154:D
5150:D
5145:i
5143:B
5131:D
5127:A
5118:i
5116:B
5111:i
5109:B
5101:A
5093:i
5091:B
5087:A
5079:A
5074:i
5072:B
5068:P
5044:)
5034:h
5030:B
5019:0
5014:0
5009:0
4992:0
4987:0
4982:0
4935:2
4931:B
4925:0
4896:1
4892:B
4883:(
4855:A
4823:)
4815:0
4805:0
4800:0
4795:1
4790:1
4783:1
4773:0
4768:0
4763:0
4758:0
4723:0
4713:1
4708:0
4703:0
4698:0
4691:0
4681:0
4676:1
4671:0
4666:0
4659:0
4649:0
4644:0
4639:1
4634:0
4626:(
4622:=
4619:M
4603:)
4601:M
4597:n
4593:n
4589:k
4585:M
4581:M
4577:A
4573:n
4569:A
4561:A
4557:m
4549:A
4545:m
4541:A
4537:A
4531:.
4528:B
4524:A
4522:r
4518:B
4509:r
4504:A
4500:r
4496:B
4492:A
4485:A
4481:A
4474:A
4472:-
4470:r
4466:r
4459:r
4447:v
4444:w
4440:A
4426:,
4423:)
4418:T
4414:w
4410:v
4407:(
4404:=
4399:i
4395:r
4390:/
4384:i
4380:A
4374:k
4371:,
4368:.
4365:.
4362:.
4359:,
4356:0
4353:=
4350:i
4342:k
4338:/
4334:1
4323:k
4301:.
4298:s
4294:2
4290:1
4288:k
4284:A
4280:A
4276:x
4273:s
4271:k
4266:2
4264:k
4259:1
4257:k
4251:x
4249:(
4247:c
4242:.
4240:A
4236:h
4232:h
4228:q
4224:d
4219:i
4217:A
4201:)
4193:d
4189:A
4178:O
4173:O
4168:O
4138:O
4128:O
4121:2
4117:A
4111:O
4104:O
4094:O
4089:O
4082:1
4078:A
4071:(
4066:=
4061:1
4054:P
4048:q
4044:A
4040:P
4030:P
4026:A
4019:e
4015:D
4011:D
4004:e
4000:B
3996:B
3988:A
3986:(
3984:ρ
3980:B
3978:(
3976:ρ
3972:A
3968:B
3961:A
3953:A
3945:A
3933:n
3929:h
3924:j
3920:A
3905:)
3897:0
3892:1
3887:1
3880:1
3875:0
3870:0
3863:1
3858:0
3853:0
3845:(
3841:=
3838:A
3813:.
3808:j
3805:i
3801:a
3795:j
3785:i
3774:r
3766:j
3763:i
3759:a
3753:j
3743:i
3703:f
3683:0
3675:i
3671:x
3650:i
3627:i
3623:x
3618:/
3612:i
3608:]
3603:x
3599:A
3596:[
3574:)
3570:x
3566:(
3563:f
3540:x
3508:O
3481:,
3476:)
3470:O
3460:O
3455:O
3450:O
3443:h
3439:A
3429:1
3423:h
3419:A
3408:O
3403:O
3398:O
3393:O
3358:O
3348:O
3341:2
3337:A
3331:O
3326:O
3319:O
3309:O
3304:O
3297:1
3293:A
3287:O
3281:(
3276:=
3271:1
3264:P
3260:A
3257:P
3231:P
3211:1
3205:h
3157:i
3153:e
3132:A
3112:A
3104:i
3100:e
3075:A
3055:h
3051:/
3044:2
3041:=
3027:.
3011:h
2991:r
2971:r
2947:h
2927:h
2907:A
2896:.
2884:r
2860:r
2839:w
2817:v
2796:A
2774:r
2750:r
2739:.
2723:A
2701:+
2696:R
2688:r
2664:r
2661:=
2658:)
2655:A
2652:(
2626:h
2606:N
2600:N
2580:A
2551:A
2546:A
2542:G
2538:A
2532:A
2526:A
2518:)
2516:A
2512:m
2503:i
2497:i
2493:A
2486:A
2482:m
2478:m
2470:A
2442:)
2439:A
2436:t
2433:(
2421:t
2399:n
2394:C
2372:)
2369:+
2366:,
2362:C
2358:(
2336:n
2331:R
2309:)
2306:+
2303:,
2299:R
2295:(
2275:F
2263:A
2259:G
2255:A
2246:a
2242:j
2238:i
2234:n
2230:n
2225:A
2221:G
2214:A
2204:G
2200:E
2183:,
2178:)
2172:G
2167:O
2160:F
2155:E
2149:(
2139:1
2132:P
2128:A
2125:P
2112:P
2105:A
2096:A
2092:n
2088:k
2083:k
2080:i
2076:e
2071:1
2068:i
2064:e
2060:F
2044:A
2031:A
2027:F
2020:n
2016:n
2012:A
1988:r
1973:h
1930:r
1909:r
1888:r
1873:)
1865:0
1860:0
1853:1
1848:0
1840:(
1836:=
1833:A
1823:r
1808:)
1800:0
1795:1
1788:1
1783:0
1775:(
1771:=
1768:A
1735:1
1732:=
1729:v
1724:T
1720:w
1709:v
1701:v
1697:w
1669:.
1664:j
1661:i
1657:a
1651:j
1641:i
1630:r
1622:j
1619:i
1615:a
1609:j
1599:i
1572:.
1567:i
1563:x
1557:i
1553:y
1547:n
1542:1
1539:=
1536:i
1527:/
1521:j
1517:x
1511:j
1508:i
1504:a
1498:i
1494:y
1488:n
1483:1
1480:=
1477:j
1474:,
1471:i
1461:z
1458:=
1455:y
1449:x
1443:,
1440:0
1434:y
1428:,
1425:0
1419:x
1406:0
1400:z
1392:=
1386:x
1377:y
1371:x
1368:A
1359:y
1350:z
1347:=
1344:y
1338:x
1332:,
1329:0
1323:y
1317:,
1314:0
1308:x
1295:0
1289:z
1281:=
1278:r
1250:.
1245:i
1241:x
1236:/
1230:i
1226:]
1222:x
1219:A
1216:[
1211:i
1207:p
1201:n
1196:1
1193:=
1190:i
1180:0
1174:x
1164:p
1156:=
1153:r
1143:x
1139:p
1131:–
1127:–
1109:.
1104:i
1100:x
1094:i
1090:y
1084:n
1079:1
1076:=
1073:i
1064:/
1058:j
1054:x
1048:j
1045:i
1041:a
1035:i
1031:y
1025:n
1020:1
1017:=
1014:j
1011:,
1008:i
998:0
992:y
982:0
976:x
968:=
962:x
953:y
947:x
944:A
935:y
926:0
920:y
910:0
904:x
896:=
890:x
881:y
875:x
872:A
863:y
854:0
848:y
838:0
832:x
824:=
821:r
811:y
807:x
799:–
790:x
782:g
778:i
773:i
769:x
764:i
759:x
757:(
755:g
751:x
744:x
736:f
731:i
729:x
725:i
720:i
716:x
711:i
706:x
704:(
702:f
698:x
687:.
681:r
665:A
649:T
645:w
641:v
638:=
633:k
629:r
624:/
618:k
614:A
602:k
582:v
577:.
550:i
546:w
542:w
534:w
530:n
526:i
521:i
517:v
505:v
501:r
497:A
492:)
489:n
485:v
481:1
478:v
474:v
467:A
463:A
459:r
451:A
443:r
438:.
432:r
418:)
415:A
412:(
396:r
392:λ
388:r
376:λ
372:A
368:r
344:r
326:n
320:j
317:,
314:i
308:1
288:0
280:j
277:i
273:a
252:n
246:n
226:)
221:j
218:i
214:a
210:(
207:=
204:A
170:A
164:(
158:A
154:k
150:A
134:A
71:(
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