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

Minimize spectral norm under diagonal similarity

Let $A$ be a real square matrix of size $n \times n$. Is there an upper bound on the minimum spectral norm under diagonal similarity, i.e., $$ s(A) = \min_{D} \lVert D^{-1} A D\rVert_2, $$ where $D$ is a non-singular, diagonal real matrix. Also, is there are a relation between $s(A)$ and the spectral radius … Read more

spectral radius monotonicity

I encountered an inequality when reading a paper. Can someone help to show how to prove it? Let be the spectral radius of matrix $A$ or $\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of matrix }A\}$. For matrices $S$ and $T$ with positive spectral radii, and two arbitrary real positive numbers $a$ and $b$, such that $\rho(S) … Read more

Find a condition such that the spectral radius for a special matrix is smaller than 1 (or a matrix norm smaller than 1)

We need a help to find a reasonable condition such that the spectral radius for a special matrix $\mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} + \mathbf{I}\otimes\mathbf{\hat{H}}$ is smaller than 1. Here $\otimes$ is tensor product. These matrices $\bf J$, $\hat{\bf{G}}$, $\hat{\bf{H}}$ are defined below. I am thinking about if there is a matrix norm $\|\|$ such that $\|\mathbf{J} \otimes\hat{\mathbf{G}}\hat{\mathbf{W}} … Read more

Projecting onto space of matrices with spectral radius less than one

Consider the space $$ S = \left\{ A \in \mathbb{R}^{n \times n} : \mathrm{SpectralRadius}(|A|) \leq 1 \right\}$$ where $|A|$ is the entry-wise absolute value. Given a matrix $M \in \mathbb{R}^{n\times n}$ such that $M \not \in S$. Is there a way to solve $$\arg\min_{\hat M \in S} \Vert M – \hat M \Vert$$ for some … 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