Let us define the following matrix multiplication:
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? Is this range of values independent of the entries in the matrix A?