Editing Perron–Frobenius theorem (section)

==Counterexamples==
The matrices ''L'' = <math>\left(
\begin{smallmatrix}
1 & 0 & 0 \\
1 & 0 & 0 \\
1 & 1 & 1
\end{smallmatrix}
\right)</math>, ''P'' = <math>\left(
\begin{smallmatrix}
1 & 0 & 0 \\
1 & 0 & 0 \\
\!-1 & 1 & 1
\end{smallmatrix}
\right)</math>, ''T'' = <math>\left(
\begin{smallmatrix}
0 & 1 & 1 \\
1 & 0 & 1 \\
1 & 1 & 0
\end{smallmatrix}
\right)</math>, ''M'' = <math>\left(
\begin{smallmatrix}
0 & 1 & 0 & 0 & 0 \\
1 & 0 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 & 0 \\
0 & 0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 & 0
\end{smallmatrix}
\right)</math> provide simple examples of what can go wrong if the necessary conditions are not met. It is easily seen that the Perron and peripheral projections of ''L'' are both equal to ''P'', thus when the original matrix is reducible the projections may lose non-negativity and there is no chance of expressing them as limits of its powers. The matrix ''T'' is an example of a primitive matrix with zero diagonal. If the diagonal of an irreducible non-negative square matrix is non-zero then the matrix must be primitive but this example demonstrates that the converse is false. ''M'' is an example of a matrix with several missing spectral teeth. If ω = e<sup>iπ/3</sup> then ω<sup>6</sup> = 1 and the eigenvalues of ''M'' are {1,ω<sup>2</sup>,ω<sup>3</sup>=-1,ω<sup>4</sup>} with a dimension 2 eigenspace for +1 so ω and ω<sup>5</sup> are both absent. More precisely, since ''M'' is block-diagonal cyclic, then the eigenvalues are {1,-1} for the first block, and {1,ω<sup>2</sup>,ω<sup>4</sup>} for the lower one{{Citation needed|date=January 2012}}