Editing Perron–Frobenius theorem (section)

====Cyclicity====
Suppose in addition that ρ(''A'') = 1 and ''A'' has ''h'' eigenvalues on the unit circle. If ''P'' is the peripheral projection then the matrix ''R'' = ''AP'' = ''PA'' is non-negative and irreducible, ''R<sup>h</sup>'' = ''P'', and the cyclic group ''P'', ''R'', ''R''<sup>2</sup>, ...., ''R''<sup>''h''−1</sup> represents the harmonics of ''A''. The spectral projection of ''A'' at the eigenvalue λ on the unit circle is given by the formula <math>\scriptstyle h^{-1}\sum^h_1\lambda^{-k}R^k</math>. All of these projections (including the Perron projection) have the same positive diagonal, moreover choosing any one of them and then taking the modulus of every entry invariably yields the Perron projection. Some donkey work is still needed in order to establish the cyclic properties (6)–(8) but it's essentially just a matter of turning the handle. The spectral decomposition of ''A'' is given by ''A''&nbsp;=&nbsp;''R''&nbsp;⊕&nbsp;(1&nbsp;−&nbsp;''P'')''A'' so the difference between ''A<sup>n</sup>'' and ''R<sup>n</sup>'' is ''A<sup>n</sup>''&nbsp;−&nbsp;''R<sup>n</sup>'' =&nbsp;(1&nbsp;−&nbsp;''P'')''A''<sup>''n''</sup> representing the transients of ''A<sup>n</sup>'' which eventually decay to zero. ''P'' may be computed as the limit of ''A<sup>nh</sup>'' as ''n''&nbsp;→&nbsp;∞.