Spectral radius of a specific matrix multiplied by a diagonal matrix $D=diag(\mathrm{e}^{i\alpha_1},...\mathrm{e}^{i\alpha_n})$

Question

Let $A=(a_{ij})\in M_n(\mathbb C)$, $n\geq 3$ be a matrix satisfying :

• $\sum_{j=1}^n\lvert a_{ij}\rvert^2=1,$ for all $i=1,\ldots n$.
• $\sum_{i=1}^n\lvert a_{ij}\rvert^2\leq1,$ for all $j=1,\ldots n$.
• $\|A\|\leq 1$ where $\|\cdot\|$ denotes the matrix norm $\|A\|=\sup_{\|v\|=1}\|Av\|\,.$

I am interested in studying the spectral radius of $D_\alpha A\,,$ where $D_\alpha$ is any diagonal matrix of the form $D_\alpha=diag(\mathrm{e}^{i\alpha_1},...\mathrm{e}^{i\alpha_n})$.

Knowing that the spectral radius of $A$ is $\rho(A)<1\,$, can we deduce that $\rho(D_\alpha A)<1\,?$

Answer 1

No. Consider the nilpotent matrix $A=\pmatrix{B&0\\ 0&B}$ where $B=\frac{1}{\sqrt{2}}\pmatrix{1&i\\ i&-1}$. Each row or column of $A$ has unit Euclidean norm. However, $\rho(DA)=\sqrt{2}>1$ when $D=\operatorname{diag}(1,-1,1,-1)$.