= Irreducible nonnegative matrix
A square <nonnegative matrix> $A$ is irreducible if for each pair $(i,j)$ there is an integer $k\geq0$ with $(A^k)_{ij}>0$. This says that every index can reach every other through positive entries. The <Perron–Frobenius theorem> then gives a positive leading <eigenvector> and an algebraically <simple eigenvalue>. Irreducibility alone permits other <eigenvalues> of the same <modulus>: the cyclic permutation <matrix> $\begin{pmatrix}0&1\\1&0\end{pmatrix}$ has <eigenvalues> $1,-1$.
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