If the absolute value of every eigenvalue of a matrix is smaller than 1, is the maximum singular value smaller than 1?

Question

I am just curious. If the absolute value of every eigenvalue of a matrix is smaller than 1, is the maximum singular value smaller than 1? This is related to my previous question Any relation between the singular values of ${\bf A}$ and ${\bf I} - {\bf A}$. Thanks!

Answer 1

For a general $\mathbf A$, this is not true. Consider $A=\begin{pmatrix}0&2\\ 0&0\end{pmatrix}$. Its eigenvalues are zero but the maximum singular value is 4. In general the spectral radius of a matrix is a lower bound on the operator norm of the matrix which means: $$ \sigma_{\max}\geq |\lambda_\max|. $$