Editing Perron–Frobenius theorem (section)

===Perron projection as a limit: ''A''<sup>''k''</sup>/''r''<sup>''k''</sup>===
Let ''A'' be a positive (or more generally, primitive) matrix, and let ''r'' be its Perron–Frobenius eigenvalue.
# There exists a limit ''A<sup>k</sup>/r<sup>k</sup>'' for ''k → ∞'', denote it by ''P''.
# ''P'' is a [[Projection (linear algebra)|projection operator]]: ''P''<sup>2</sup> = ''P'', which commutes with ''A'': ''AP'' = ''PA''.
# The image of ''P'' is one-dimensional and spanned by the Perron–Frobenius eigenvector ''v'' (respectively for ''P<sup>T</sup>''—by the Perron–Frobenius eigenvector ''w'' for ''A<sup>T</sup>'').
# ''P'' = ''vw''<sup>''T''</sup>, where ''v,w'' are normalized such that ''w''<sup>''T''</sup> ''v'' = 1.
# Hence ''P'' is a positive operator.

Hence ''P'' is a [[spectral projection]] for the Perron–Frobenius eigenvalue ''r'', and is called the Perron projection. The above assertion is not true for general non-negative irreducible matrices.

Actually the claims above (except claim 5) are valid for any matrix ''M'' such that there exists an eigenvalue ''r'' which is strictly greater than the other eigenvalues in absolute value and is the simple root of the characteristic [[polynomial]]. (These requirements hold for primitive matrices as above).

Given that ''M'' is diagonalizable, ''M'' is conjugate to a diagonal matrix with eigenvalues ''r''<sub>1</sub>, ... , ''r''<sub>''n''</sub> on the diagonal (denote ''r''<sub>1</sub> = ''r''). The matrix ''M''<sup>''k''</sup>/''r''<sup>''k''</sup> will be conjugate (1, (''r''<sub>2</sub>/''r'')<sup>''k''</sup>, ... , (''r''<sub>''n''</sub>/''r'')<sup>''k''</sup>), which tends to (1,0,0,...,0), for ''k → ∞'', so the limit exists. The same method works for general ''M'' (without assuming that ''M'' is diagonalizable).

The projection and commutativity properties are elementary corollaries of the definition: ''MM''<sup>''k''</sup>/''r''<sup>''k''</sup> =  ''M''<sup>''k''</sup>/''r''<sup>''k''</sup> ''M'' ; ''P''<sup>2</sup> = lim ''M''<sup>2''k''</sup>/''r''<sup>2''k''</sup> = ''P''. The third fact is also elementary: ''M''(''Pu'') = ''M'' lim ''M''<sup>''k''</sup>/''r''<sup>''k''</sup> ''u'' = lim ''rM''<sup>''k''+1</sup>/''r''<sup>''k''+1</sup>''u'', so taking the limit yields ''M''(''Pu'') = ''r''(''Pu''), so image of ''P'' lies in the ''r''-eigenspace for ''M'', which is one-dimensional by the assumptions.

Denoting by ''v'', ''r''-eigenvector for ''M'' (by ''w'' for ''M<sup>T</sup>''). Columns of ''P'' are multiples of ''v'', because the image of ''P'' is spanned by it. Respectively, rows of ''w''. So ''P'' takes a form ''(a v w<sup>T</sup>)'', for some ''a''. Hence its trace equals to ''(a w<sup>T</sup> v)''. Trace of projector equals the dimension of its image. It was proved before that it is not more than one-dimensional. From the definition one sees that ''P'' acts identically on the ''r''-eigenvector for ''M''. So it is one-dimensional. So choosing (''w''<sup>''T''</sup>''v'') = 1, implies ''P'' = ''vw''<sup>''T''</sup>.