Perron–Frobenius theorem | Wikipedia audio article

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00:00:51 1 Statement
00:01:42 1.1 Positive matrices
00:02:42 1.2 Non-negative matrices
00:06:33 1.2.1 Classification of matrices
00:08:18 1.2.2 Perron–Frobenius theorem for irreducible matrices
00:11:21 1.3 Further properties
00:14:40 2 Applications
00:17:45 2.1 Non-negative matrices
00:18:07 2.2 Stochastic matrices
00:20:16 2.3 Algebraic graph theory
00:21:00 2.4 Finite Markov chains
00:21:36 2.5 Compact operators
00:22:02 3 Proof methods
00:22:47 3.1 Perron root is strictly maximal eigenvalue for positive (and primitive) matrices
00:23:14 3.1.1 Proof for positive matrices
00:23:49 3.1.2 Lemma
00:24:02 3.2 Power method and the positive eigenpair
00:24:39 3.3 Multiplicity one
00:25:23 3.4 No other non-negative eigenvectors
00:25:54 3.5 Collatz–Wielandt formula
00:28:09 3.6 Perron projection as a limit: iAsupk/sup/rsupk/sup/i
00:31:14 3.7 Inequalities for Perron–Frobenius eigenvalue
00:31:58 3.8 Further proofs
00:32:46 3.8.1 Perron projection
00:32:56 3.8.2 Peripheral projection
00:33:20 3.8.3 Cyclicity
00:34:03 4 Caveats
00:34:16 5 Terminology
00:34:40 6 See also
00:35:02 7 Notes
00:36:18 8 References
00:36:31 8.1 Original papers
00:37:27 8.2 Further reading
00:37:39 Inequalities for Perron–Frobenius eigenvalue
00:40:48 rw and the smallest component of w (say wi) is 1. Then r
00:41:16 (1, 1, ..., 1) and immediately obtains the inequality.
00:41:23 Further proofs
00:41:32 ==== Perron projection
00:42:17 PA
00:42:34 λx then PAx
00:42:45 ρ(A)Px which means Px
00:43:04 1 then it can be decomposed as P ⊕ (1 − P)A so that An
00:43:53 Peripheral projection
00:44:17 1. So we consider the peripheral projection, which is the spectral projection of A corresponding to all the eigenvalues that have modulus ρ(A). 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.
00:44:34 Cyclicity
00:44:42 Suppose in addition that ρ(A)
00:44:58 AP
00:46:03 R ⊕ (1 − P)A so the difference between An and Rn is An − Rn
00:46:26 Caveats
00:47:54 eiπ/3 then ω6
00:48:12 Terminology
00:49:21 See also

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SUMMARY
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In linear algebra, the Perron–Frobenius theorem, proved by Oskar Perron (1907) and Georg Frobenius (1912), asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type); to economics (Okishio's theorem, Hawkins–Simon condition);
to demography (Leslie population age distribution model);
to social networks (DeGroot learning process), to Internet search engines and even to ranking of football
teams. The first to discuss the ordering of players within tournaments using Perron–Frobenius eigenvectors is Edmund Landau.