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I have a matrix $A \in \mathbb{R}^{n \times n}$ such that its elements are all non-negative values.

I know that for any $k$, $A^k$ has elements on the diagonal which are smaller or equal to 1.

Can I show that the largest eigenvalue of $A$ is smaller than 1? I am pretty sure that's true, but I am not completely sure.

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up vote 2 down vote accepted

The largest eigenvalue could be equal to $1$ (e.g. if $A$ is upper triangular with at least one $1$ on the diagonal). It could be complex with absolute value $1$ (e.g. for a permutation matrix). But it can't be greater than $1$ in absolute value. This follows from the Perron-Frobenius theorem. First, replacing $A$ by $A^p$ if necessary, we can assume $A$ is aperiodic. Next, assume $A$ is irreducible (otherwise look at each irreducible component separately). If $\lambda$ is the Perron eigenvalue, there are positive vectors $u$ and $v$ such that $A^n - \lambda^n u v^T = o(\lambda^n)$. In particular, if $\lambda > 1$ every entry of $A^n$ goes to $+\infty$ as $n \to \infty$.

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thank you very much! – kloop Sep 30 '12 at 3:52
something is strange here: even if the diagonal elements of $A^k$ are larger than 1, but bounded, your proof still goes through. I find it quite odd. (Though, maybe it is impossible to have $A^k$ on the diagonal larger than 1, but bounded for all $k$, so it might be a vacuous statement.) – kloop Sep 30 '12 at 16:45
also, what kind of constraint on the diagonal of $A$ might be required to show that the largest eigenvalue is strictly smaller than 1? What if the diagonal elements are strictly smaller than 1? – kloop Sep 30 '12 at 17:21
For a nonnegative matrix $A$, $(A^n)_{jj} \ge (A_{jj})^n$. So if $A$ has a diagonal element $> 1$, that diagonal element in $A^n$ goes to $+\infty$ as $n \to \infty$. – Robert Israel Sep 30 '12 at 18:48
The largest eigenvalue (in absolute value) is strictly less than $1$ iff the diagonal entries of $A^n$ all go to $0$ as $n \to \infty$. – Robert Israel Sep 30 '12 at 18:52

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