I have a matrix A and a matrix B such that $B_{i, j} = |A_{i, j}|$. I am given that all of the eigenvalues of B have magnitude less than 1, and therefore: $ \displaystyle \lim_{k \to \infty} B^k = 0$ through analysis of the Jordan normal form.
What I want to prove now is that $\displaystyle \lim_{k \to \infty} A^k = 0$ and that, therefore, all the eigenvalues of A also have magnitude less than 1. At first, this seems intuitively true, but I'm struggling to find a formal proof for the convergence of $A^k$. Can someone offer a formal explanation?
Hint: Try and show (e.g. by induction on $k$) that for all $i,j$, we have $\left|A^{k}_{i,j}\right| \leq B^{k}_{i,j}$ for any $k\in \mathbb{Z}^{+}$. You can use the triangle inequality and definition of matrix multiplication. ($A^k_{i,j}$ refers to the $i,j$ element of $A^{k}$, and similarly for $B$.)