Editing Perron–Frobenius theorem (section)

===Collatz–Wielandt formula===
Given a positive (or more generally irreducible non-negative matrix) ''A'', one defines 
the function ''f'' on the set of all non-negative non-zero vectors ''x'' such that ''f(x)'' is 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]] is the Perron–Frobenius eigenvalue ''r''.

For the proof we denote the maximum of ''f'' by the value ''R''. The proof requires to show '' R = r''. Inserting the Perron-Frobenius eigenvector ''v'' into ''f'', we obtain ''f(v) = r'' and conclude ''r ≤ R''. For the opposite inequality, we consider an arbitrary nonnegative vector ''x'' and let ''ξ=f(x)''. The definition of ''f'' gives ''0 ≤ ξx ≤ Ax'' (componentwise). Now, we use the positive right eigenvector ''w'' for ''A'' for the Perron-Frobenius eigenvalue ''r'', then '' ξ w<sup>T</sup> x = w<sup>T</sup> ξx ≤ w<sup>T</sup> (Ax) = (w<sup>T</sup> A)x = r w<sup>T</sup> x ''. Hence ''f(x) = ξ ≤ r'', which implies 
''R ≤ r''.<ref>{{harvnb|Meyer|2000|pp=[http://www.matrixanalysis.com/Chapter8.pdf chapter 8 page 666] {{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>