Let us define the following matrix multiplication:
$$C=AB$$
where $B$ is a block diagonal matrix with $N$ blocks, $B_1$, $B_2$,…, $B_N$, each of dimensions $M \times M$. I know that $B_k = I_M – \mu R_k$ with $R_k$ equals to a hermitian matrix and $\mu$ equal to some positive constant. Moreover, I know that the the entries of the matrix $A$ are non-negative real numbers. I also know that the matrix $A$ is right stochastic, i.e., the sum of the elements in each row equals one. Can I say that the spectral radius of C is smaller than one for some values of $\mu$? If so, can I determine the range of values of $\mu$ under which the spectral radius of $C$ is smaller than one?
Best Answer
The answer is no in general. Suppose $A$ is stochastic but not doubly stochastic. Then its operator norm (i.e. the largest singular value) is strictly greater than $1$. Let $A=USV^T$ be its singular value decomposition. Let $0<\frac1{\|A\|_2}<\epsilon<1$ and define $B=\epsilon VU^T$. Then $B$ itself is a big $n\times n$ block matrix whose operator norm is equal to $\epsilon<1$. Yet the spectral radius of $AB=\epsilon USU^T$ is $\epsilon\|A\|_2$, which is greater than $1$.
By extending the above construction, we can construct counterexamples such that both $A$ and $B$ are block diagonal and they have identical partitions. Therefore, by continuity, there exist counterexample where the $A$ is entrywise nonzero but $B$ is genuinely/properly block-diagonal as well.