Editing Perron–Frobenius theorem (section)

==Applications==
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.

===Non-negative matrices===
The Perron–Frobenius theorem does not apply directly to non-negative matrices. Nevertheless, any reducible square matrix ''A'' may be written in upper-triangular block form (known as the '''normal form of a reducible matrix''')<ref>{{harvnb|Varga|2002|p=2.43 (page 51)}}</ref>
::::''PAP''<sup>−1</sup> = <math> \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)</math>

where ''P'' is a permutation matrix and each ''B<sub>i</sub>'' is a square matrix that is either irreducible or zero. Now if ''A'' is
non-negative then so too is each block of ''PAP''<sup>−1</sup>, moreover the spectrum of ''A'' is just the union of the spectra of the
''B<sub>i</sub>''.

The invertibility of ''A'' can also be studied. The inverse of ''PAP''<sup>−1</sup> (if it exists) must have diagonal blocks of the form ''B<sub>i</sub>''<sup>−1</sup> so if any
''B<sub>i</sub>'' isn't invertible then neither is ''PAP''<sup>−1</sup> or ''A''.
Conversely let ''D'' be the block-diagonal matrix corresponding to ''PAP''<sup>−1</sup>, in other words ''PAP''<sup>−1</sup> with the
asterisks zeroised. If each ''B<sub>i</sub>'' is invertible then so is ''D'' and ''D''<sup>−1</sup>(''PAP''<sup>−1</sup>) is equal to the
identity plus a nilpotent matrix. But such a matrix is always invertible (if ''N<sup>k</sup>'' = 0 the inverse of 1 − ''N'' is
1 + ''N'' + ''N''<sup>2</sup> + ... + ''N''<sup>''k''−1</sup>) so ''PAP''<sup>−1</sup> and ''A'' are both invertible.

Therefore, many of the spectral properties of ''A'' may be deduced by applying the theorem to the irreducible ''B<sub>i</sub>''. For example, the Perron root is the maximum of the ρ(''B<sub>i</sub>''). While there will still be eigenvectors with non-negative components it is quite possible
that none of these will be positive.

===Stochastic matrices===
A row (column) [[stochastic matrix]] 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.

If ''A'' is row-stochastic then the column vector with each entry 1 is an eigenvector corresponding to the eigenvalue 1, which is also ρ(''A'') by the remark above. It might not be the only eigenvalue on the unit circle: and the associated eigenspace can be multi-dimensional. If ''A'' is row-stochastic and irreducible then the Perron projection is also row-stochastic and all its rows are equal.

===Algebraic graph theory===
The theorem has particular use in [[algebraic graph theory]]. The "underlying graph" of a nonnegative ''n''-square matrix is the graph with vertices numbered 1, ..., ''n'' and arc ''ij'' if and only if ''A<sub>ij</sub>'' ≠ 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 [[adjacency matrix]] of a [[strongly connected component|strongly connected graph]] is irreducible.<ref>{{cite book |author-link=Richard A. Brualdi |first1=Richard A. |last1=Brualdi |author-link2=H. J. Ryser |first2=Herbert J. |last2=Ryser |title=Combinatorial Matrix Theory |url=https://archive.org/details/combinatorialmat0000brua_x9u3 |url-access=registration |location=Cambridge |publisher=Cambridge UP |year=1992 |isbn=978-0-521-32265-2 }}</ref><ref>{{cite book |author-link=Richard A. Brualdi |first1=Richard A. |last1=Brualdi |first2=Dragos |last2=Cvetkovic |title=A Combinatorial Approach to Matrix Theory and Its Applications |publisher=CRC Press |location=Boca Raton, FL |year=2009 |isbn=978-1-4200-8223-4 }}</ref>

===Finite Markov chains===
The theorem has a natural interpretation in the theory of finite [[Markov chain]]s (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 [[subshift of finite type]]).<!--which article?-->

===Compact operators===
{{main|Krein–Rutman theorem}}
More generally, it can be extended to the case of non-negative [[compact operator]]s, which, in many ways, resemble finite-dimensional matrices. These are commonly studied in physics, under the name of [[transfer operator]]s, or sometimes '''Ruelle–Perron–Frobenius operators''' (after [[David Ruelle]]). In this case, the leading eigenvalue corresponds to the [[thermodynamic equilibrium]] of a [[dynamical system]], 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 [[arrow of time]] in what would otherwise appear to be reversible, deterministic dynamical processes, when examined from the point of view of [[point-set topology]].<ref>{{cite book |first=Michael C. |last=Mackey |title=Time's Arrow: The origins of thermodynamic behaviour |location=New York |publisher=Springer-Verlag |year=1992 |isbn=978-0-387-97702-7 }}</ref>