Suppose $A$ is an $n\times n$ matrix over the set of non-negative integers. Is there a necessary condition for $A$ so that it would have at least one real non-zero eigenvalue?
Ignore what's written below:
I'm just wondering, given an $n\times n$ matrix $A$ over the set of non-negative integers, is there a necessary condition for $A$ to have at least one real, non-zero eigenvalue?
Edit: If we know that $A$ doesn't have a row that is completely zero, would this suffice?
Best Answer
By the Perron-Frobenius theorem and its extension to the nonnegative matrices, a matrix with nonnegative elements either has only zeroes as the eigenvalues or it has to have at least one real non-zero eigenvalue, because it has one that is - in absolute value - greater than or equal to all the others (so it cannot be zero).