## 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 … Read more

## The spectral radius of a n×nn\times n matrix

I would like to know which is the spectral radius of this n×n matrix: 01…110…0…………10…0 I know that the spectral radius is the maximum eigenvalue, but I don’t know how to calculate it in this matrix… I also know that if we’ve got a symmetric amtrix the spectral radius is ||A||2 but I neither know … Read more

## Prove using Schur decomposition lim\lim_{n \rightarrow \infty}||A^n||=0 \iff p(A)<1

a) Prove using Schur decomposition \lim_{n \rightarrow \infty}||A^n||=0 \iff p(A)<1 where A is in \mathbb{C}^{mxm} and p is the spectral radius. b) \lim_{t \rightarrow \infty}||e^{At}||=0 \iff a(A)<1 where A is in \mathbb{C}^{mxm} and a is the spectral abscissa. My thoughts are as follows. For (a), I think that \lim_{n \rightarrow \infty}||A^n||=0 will only happen if … Read more

## Matrix norm for two matrices simultaneously close to spectral radius

Suppose A and B have the same spectral radius ρ. We can find a norm ‖ s.t. \|A\|_A – \epsilon < \rho. We can likewise find a another norm s.t. \|B\|_B – \epsilon < \rho. Under what conditions can we find a norm \|\cdot\|_* such that both \|A\|_* – \epsilon < \rho and \|B\|_* – … Read more