Editing Perron–Frobenius theorem (section)

===No other non-negative eigenvectors===
Given positive (or more generally irreducible non-negative matrix) ''A'', the Perron–Frobenius eigenvector is the only (up to multiplication by constant) non-negative eigenvector for ''A''.

Other eigenvectors must contain negative or complex components since eigenvectors for different eigenvalues are orthogonal in some sense, but two positive eigenvectors cannot be orthogonal, so they must correspond to the same eigenvalue, but the eigenspace for the Perron–Frobenius is one-dimensional.

Assuming there exists an eigenpair (''λ'', ''y'') for ''A'', such that vector ''y'' is positive, and given (''r'', ''x''), where ''x'' – is the left Perron–Frobenius eigenvector for ''A'' (i.e. eigenvector for ''A<sup>T</sup>''), then
''rx''<sup>''T''</sup>''y'' = (''x''<sup>''T''</sup> ''A'') ''y'' = ''x''<sup>''T''</sup> (''Ay'') = ''λx''<sup>''T''</sup>''y'', also ''x''<sup>''T''</sup> ''y'' > 0, so one has: ''r'' = ''λ''. Since the eigenspace for the Perron–Frobenius eigenvalue ''r'' is one-dimensional, non-negative eigenvector ''y'' is a multiple of the Perron–Frobenius one.<ref>{{harvnb|Meyer|2000|pp=[http://www.matrixanalysis.com/Chapter8.pdf chapter 8 claim 8.2.10 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>