# Spectral radius of the product of a right stochastic matrix and a block diagonal matrix

Let us define the following matrix:

$C=AB$

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 $\mu$ 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. In particular, the matrix has the following structure

$A=blkdiag\{A_g \otimes I_{M-1},I_N\}$

where $\otimes$ denotes the Kronecker product, $I_{M-1}$ is an $(M-1 \times M-1)$ identity matrix, $I_N$ equals an $(N\times N)$ identity matrix, $A_g$ denotes a $(N\times N)$ right stochastic matrix with non-negative real entries and $blkdiag\{.\}$ equals a block diagonal matrix.

Would all this information help to get a result independent of the dimensions of $A$, i.e., $MN$? 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?

• By "matrix 2-norm" do you mean the $\ell^2$ norm of the matrix entries (i.e. the Hilbert-Schmidt norm) or the operator norm induced by the $\ell^2$ norm on vectors (i.e. the spectral norm)? Feb 20, 2016 at 13:27
• I have already clarified the question. Sorry for the bad explanation before. Please check the new statement. If there is still something unclear, please let me know Feb 20, 2016 at 13:38
• If $R_k$ is Hermitian, so is $\mu R_k$. Do you know more about the $R_k$ (e.g., positivity, bounded spectral radius)? Feb 20, 2016 at 15:49
• Yes, $R_k$ is positive definite. Feb 20, 2016 at 18:10