Editing Perron–Frobenius theorem (section)

===Positive matrices===
Let <math>A = (a_{ij}) </math> be an <math> n \times n </math> positive matrix: <math> a_{ij} > 0 </math> for <math> 1 \le i,j \le n </math>. Then the following statements hold.

# There is a positive real number ''r'', called the '''Perron root''' or the '''Perron–Frobenius eigenvalue''' (also called the '''leading eigenvalue''', '''principal eigenvalue''' or '''dominant eigenvalue'''), such that ''r'' is an eigenvalue of ''A'' and any other eigenvalue ''λ'' (possibly [[complex number|complex]]) in [[absolute value]] is strictly smaller than ''r'' , |''λ''| < ''r''. Thus, the [[spectral radius]] <math>\rho(A) </math> is equal to ''r''. If the matrix coefficients are algebraic, this implies that the eigenvalue is a [[Perron number]].
# The Perron–Frobenius eigenvalue is simple: ''r'' is a simple root of the [[characteristic polynomial]] of ''A''. Consequently, the [[eigenspace]] associated to ''r'' is one-dimensional. (The same is true for the left eigenspace, i.e., the eigenspace for ''A<sup>T</sup>'', the transpose of ''A''.)
# There exists an eigenvector ''v'' = (''v''<sub>1</sub>,...,''v''<sub>''n''</sub>)''<sup>T</sup>'' of ''A'' with eigenvalue ''r'' such that all components of ''v'' are positive: ''A v'' = ''r v'', ''v''<sub>''i''</sub> > 0 for 1 ≤ ''i'' ≤ ''n''. (Respectively, there exists a positive left eigenvector ''w'' : ''w<sup>T</sup> A'' = ''w<sup>T</sup>'' r, ''w''<sub>''i''</sub> > 0.) It is known in the literature under many variations as the '''Perron vector''', '''Perron eigenvector''', '''Perron-Frobenius eigenvector''', '''leading eigenvector''', '''principal eigenvector''' or '''dominant eigenvector'''. 
# There are no other positive (moreover non-negative) eigenvectors except positive multiples of ''v'' (respectively, left eigenvectors except ww'w''), i.e., all other eigenvectors must have at least one negative or non-real component.
# <math> \lim_{k \rightarrow \infty} A^k/r^k = 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 ''vw<sup>T</sup>'' is the [[Jordan canonical form#Invariant subspace decompositions|projection onto the eigenspace]] corresponding to&nbsp;''r''. This projection is called the '''Perron projection'''.
# '''[[Lothar Collatz|Collatz]]–Wielandt formula''': for all non-negative non-zero vectors ''x'', let ''f''(''x'') be the minimum value of [''Ax'']<sub>''i''</sub> / ''x''<sub>''i''</sub> taken over all those ''i'' such that ''x<sub>i</sub>'' ≠ 0. Then ''f'' is a real valued function whose [[maximum]] over all non-negative non-zero vectors ''x'' is the Perron–Frobenius eigenvalue.
# A "Min-max" Collatz–Wielandt formula takes a form similar to the one above: for all strictly positive vectors ''x'', let ''g''(''x'') be the maximum value of [''Ax'']<sub>''i''</sub> / ''x''<sub>''i''</sub> taken over ''i''. Then ''g'' is a real valued function whose [[minimum]] over all strictly positive vectors ''x'' is the Perron–Frobenius eigenvalue.
# '''[[Garrett Birkhoff|Birkhoff]]–[[Richard S. Varga|Varga]] formula''': Let ''x'' and ''y'' be strictly positive vectors.  Then,<ref>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.</ref><math display="block">r = \sup_{x>0} \inf_{y>0} \frac{y^\top A x}{y^\top x} = \inf_{x>0} \sup_{y>0} \frac{y^\top A x}{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.</math> 
# '''[[Monroe D. Donsker|Donsker]]–[[S. R. Srinivasa Varadhan|Varadhan]]–[[Shmuel Friedland|Friedland]] formula''': Let ''p'' be a probability vector and ''x'' a strictly positive vector.  Then,<ref>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.</ref><ref>Friedland, S., 1981. Convex spectral functions. Linear and multilinear algebra, 9(4), pp.299-316.</ref> <math display="block">r = \sup_p \inf_{x>0} \sum_{i=1}^n p_i[Ax]_i/x_i.</math> 
# '''[[Miroslav Fiedler|Fiedler]] formula''':<ref>{{cite journal |author1=Miroslav Fiedler |author2=Charles R. Johnson |author3=Thomas L. Markham |author4=Michael Neumann |title=A Trace Inequality for M-matrices and the Symmetrizability of a Real Matrix by a Positive Diagonal Matrix |journal=Linear Algebra and Its Applications |date=1985 |volume=71 |pages=81–94 |doi=10.1016/0024-3795(85)90237-X |doi-access=free }}</ref> <math display="block">r = \sup_{z > 0} \ \inf_{x>0, \ y>0,\ x \circ y = z} \frac{y^\top A x}{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.</math>
# The Perron–Frobenius eigenvalue satisfies the inequalities <math display="block">\min_i \sum_{j} a_{ij} \le r \le \max_i \sum_{j} a_{ij}.</math>

All of these properties extend beyond strictly positive matrices to '''primitive matrices''' (see below).  Facts 1–7 can be found in Meyer<ref name="Meyer"/> [https://web.archive.org/web/20100307021652/http://www.matrixanalysis.com/Chapter8.pdf chapter 8] claims 8.2.11–15 page 667 and exercises 8.2.5,7,9 pages 668–669.

The left and right eigenvectors ''w'' and ''v'' are sometimes normalized so that the sum of their components is equal to 1; in this case, they are sometimes called '''stochastic eigenvectors'''.  Often they are normalized so that the right eigenvector ''v'' sums to one, while <math>w^T v=1</math>.