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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? Is this range of values independent of the entries in the matrix A?

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Let $\|\cdot\|$ denote the spectral norm. Because $A$ is stochastic, its spectral norm is bounded above by $MN$ (the sum of the absolute value of all entries). Because $B$ is Hermitian, $\|B\| = \rho(B)$, the spectral radius of $B$. We then have $$ \|C\| = \|AB\| \leq \|A\|\,\|B\| \leq MN \rho(B) $$ Thus, if $\mu$ is such that $\rho(B) < \frac 1{MN}$, then we can guarantee that $\rho(C) \leq \|C\| < 1$ independently of the entries of $A$.

If we had more information about $A$, we might have an easier time finding such values of $\mu$.

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  • $\begingroup$ What about if $A=blkdiag\{A_g \otimes I_{M-1},I_N\}$ with $\otimes$ denoting the Kronecker product, $I_{M-1}$ an $(M-1 \times M-1)$ identity matrix, $I_N$ equal to an $(N\times N)$ identity matrix, $A_g$ denoting a $(N\times N)$ right stochastic matrix with non-negative real entries and $blkdiag\{.\}$ denoting a block diagonal matrix? Would this information help to get a result independent of the dimensions of $A$, i.e., $MN$? $\endgroup$
    – user316165
    Feb 20 '16 at 19:58

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