Suppose I have one $pN\times pN$ matrix $\bf A$ with spectral radius no larger than 1 (maximum of absolute values of eigenvalues is no larger than 1), and the other matrix $\bf H$ is in a block-like format (empty means zero, only zeros in the top-left and bottom-right block are explicitly marked, the superscript like $N^{(N+1,N)}$ means this number "$N$" is at the $N+1$th row and $N$th column)
My question is how to derive a reasonably tight bound of the spectral radius of the sum $\bf A+H$. Again the spectral radius of $\bf A$ is smaller than 1. The eigenvalue of $\bf H$ is $\pm \frac{1}{{n + 1}}$, so we believe the spectral radius of $\bf{A+H}$ should be near the spectral radius of $\bf A$.