# 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)? – Ian Morris Feb 20 '16 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 – user87933 Feb 20 '16 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)? – Sebastian Goette Feb 20 '16 at 15:49
• Yes, $R_k$ is positive definite. – user87933 Feb 20 '16 at 18:10