= Perron–Frobenius theorem
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{title2=$|\lambda_j|<\lambda_1\quad(j>1)$}
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The strictly positive case states that a real square <matrix> $W$ with $W_{ij}>0$ has a positive <simple eigenvalue> $\lambda$ and positive left and right <eigenvectors>, with all other <eigenvalues> strictly smaller in <modulus>. The <Brouwer fixed-point theorem> applied to $v\mapsto Wv/(\mathbf1^TWv)$ on the nonnegative unit <simplex> gives $v>0$ and $Wv=\lambda v$. For another <eigenvector> $z$, maximize $|z_i|/v_i$; the <triangle inequality> gives $|\mu|\leq\lambda$. Equality forces every <modulus> ratio and <complex argument> to agree because every entry is positive, so $z$ is proportional to $v$ and $\mu=\lambda$. Apply the same argument to $W^T$ for a positive left <eigenvector>; its positive pairing with $v$ rules out a <generalized eigenvector> at $\lambda$, establishing algebraic simplicity. For merely nonnegative matrices a nonnegative leading <eigenvector> exists. An <irreducible nonnegative matrix> has a positive leading <eigenvector> and simple Perron <eigenvalue>; a <primitive nonnegative matrix> has the strict <modulus> gap.
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