Editing Perron–Frobenius theorem (section)

===Further properties===
Let ''A'' be an irreducible non-negative matrix, then:
# (I+''A'')<sup>''n''−1</sup> is a positive matrix. (Meyer<ref name="Meyer"/> [https://web.archive.org/web/20100307021652/http://www.matrixanalysis.com/Chapter8.pdf claim 8.3.5 p. 672]). For a non-negative ''A'', this is also a sufficient condition.<ref name="Minc">{{cite book |first=Henryk |last= Minc |author-link=Henryk Minc |title=Nonnegative matrices |isbn=0-471-83966-3 |year=1988 |publisher=John Wiley & Sons |location=New York |page=6 [Corollary 2.2] }}</ref>
# Wielandt's theorem.<ref>{{Cite book |author=Gradshtein, Izrailʹ Solomonovich |url=http://worldcat.org/oclc/922964628 |title=Table of integrals, series, and products |date=18 September 2014 |publisher=Elsevier |isbn=978-0-12-384934-2 |oclc=922964628}}</ref>{{clarify|reason=What are the restrictions on ''B''?|date=March 2015}} If |''B''|<''A'', then ''ρ''(''B'')≤''ρ''(''A''). If equality holds (i.e. if ''μ=ρ(A)e<sup>iφ</sup>'' is eigenvalue for ''B''), then ''B'' = ''e''<sup>''iφ''</sup> ''D AD''<sup>−1</sup> for some diagonal unitary matrix ''D'' (i.e. diagonal elements of ''D'' equals to ''e''<sup>''iΘ''<sub>''l''</sub></sup>, non-diagonal are zero).<ref name="Meyer675">{{harvnb|Meyer|2000|pp=[http://www.matrixanalysis.com/Chapter8.pdf claim 8.3.11 p. 675] {{cite web |url=http://www.matrixanalysis.com/Chapter8.pdf |title=Archived copy |access-date=2010-03-07 |url-status=dead |archive-url=https://web.archive.org/web/20100307021652/http://www.matrixanalysis.com/Chapter8.pdf |archive-date=March 7, 2010 }}}}</ref>
# If some power ''A<sup>q</sup>'' is reducible, then it is completely reducible, i.e. for some permutation matrix ''P'', it is true that: <math>
P A^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}
</math>, where ''A<sub>i</sub>'' are irreducible matrices having the same maximal eigenvalue. The number of these matrices ''d'' is the greatest common divisor of ''q'' and ''h'', where ''h'' is period of ''A''.<ref>{{harvnb|Gantmacher|2000|p=section XIII.5 theorem 9}}</ref>
# If ''c''(''x'') ''= x<sup>n</sup> + c<sub>k<sub>1</sub></sub> x<sup>n-k<sub>1</sub></sup> + c<sub>k<sub>2</sub></sub> x<sup>n-k<sub>2</sub></sup> + ... + c<sub>k<sub>s</sub></sub> x<sup>n-k<sub>s</sub></sup>'' is the characteristic polynomial of ''A'' in which only the non-zero terms are listed, then the period of ''A'' equals the greatest common divisor of ''k<sub>1</sub>, k<sub>2</sub>, ... , k<sub>s</sub>''.<ref>{{harvnb|Meyer|2000|pp=[http://www.matrixanalysis.com/Chapter8.pdf page 679] {{cite web |url=http://www.matrixanalysis.com/Chapter8.pdf |title=Archived copy |access-date=2010-03-07 |url-status=dead |archive-url=https://web.archive.org/web/20100307021652/http://www.matrixanalysis.com/Chapter8.pdf |archive-date=March 7, 2010 }}}}</ref>
# [[Cesàro summation|Cesàro]] [[summability theory|averages]]: <math> \lim_{k \rightarrow \infty} 1/k\sum_{i=0,...,k} A^i/r^i = ( v w^T),</math> where the left and right eigenvectors for ''A'' are normalized so that ''w''<sup>''T''</sup>''v'' = 1. Moreover, the matrix ''v w<sup>T</sup>'' is the [[Spectral theorem|spectral projection]] corresponding to ''r'', the Perron projection.<ref>{{harvnb|Meyer|2000|pp=[http://www.matrixanalysis.com/Chapter8.pdf example 8.3.2 p. 677] {{cite web |url=http://www.matrixanalysis.com/Chapter8.pdf |title=Archived copy |access-date=2010-03-07 |url-status=dead |archive-url=https://web.archive.org/web/20100307021652/http://www.matrixanalysis.com/Chapter8.pdf |archive-date=March 7, 2010 }}}}</ref>
# Let ''r'' be the Perron–Frobenius eigenvalue, then the adjoint matrix for (''r''-''A'') is positive.<ref>{{harvnb|Gantmacher|2000|p=[https://books.google.com/books?id=cyX32q8ZP5cC&q=preceding%20section&pg=PA62 section XIII.2.2 page 62]}}</ref>
# If ''A'' has at least one non-zero diagonal element, then ''A'' is primitive.<ref>{{harvnb|Meyer|2000|pp= [http://www.matrixanalysis.com/Chapter8.pdf example 8.3.3 p. 678] {{cite web |url=http://www.matrixanalysis.com/Chapter8.pdf |title=Archived copy |access-date=2010-03-07 |url-status=dead |archive-url=https://web.archive.org/web/20100307021652/http://www.matrixanalysis.com/Chapter8.pdf |archive-date=March 7, 2010 }}}}</ref>
# If 0 ≤ ''A'' < ''B'', then ''r''<sub>''A''</sub> ≤ ''r''<sub>''B.''</sub> Moreover, if ''B'' is irreducible, then the inequality is strict: ''r<sub>A</sub> < r<sub>B</sub>''.

A matrix ''A'' is primitive provided it is non-negative and ''A<sup>m</sup>'' is positive for some ''m'', and hence ''A<sup>k</sup>'' is positive for all ''k ≥ m''. To check primitivity, one needs a bound on how large the minimal such ''m'' can be, depending on the size of ''A'':<ref>{{harvnb|Meyer|2000|pp=[http://www.matrixanalysis.com/Chapter8.pdf chapter 8 example 8.3.4 page 679 and exercise 8.3.9 p.&nbsp;685] {{cite web |url=http://www.matrixanalysis.com/Chapter8.pdf |title=Archived copy |access-date=2010-03-07 |url-status=dead |archive-url=https://web.archive.org/web/20100307021652/http://www.matrixanalysis.com/Chapter8.pdf |archive-date=March 7, 2010 }}}}</ref>
* If ''A'' is a non-negative primitive matrix of size ''n'', then ''A''<sup>''n''<sup>2</sup>&nbsp;−&nbsp;2''n''&nbsp;+&nbsp;2</sup> is positive. Moreover, this is the best possible result, since for the matrix ''M'' below, the power ''M<sup>k</sup>'' is not positive for every ''k'' < ''n''<sup>2</sup>&nbsp;−&nbsp;2''n''&nbsp;+&nbsp;2, since (''M''<sup>''n''<sup>2</sup>&nbsp;−&nbsp;2''n''+1</sup>)<sub>1,1</sub> = 0.
:<math>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)
</math>