For $ A = (A_1,\cdots,A_d)\in\mathcal{L}(E)^d$ , the algebraic spectral radius of $ A$ was given by $ $ r_a(A)=\lim_{n\to+\infty}\left\|\sum_{f\in F(n,d)} A_f^* A_f\right\|^{\frac{1}{2n}} , $ $ where $ F(n,d):=\{f:\,\{1,\cdots,n\}\longrightarrow \{1,\cdots,d\}\}$ and $ A_f:=A_{f(1)}\cdots A_{f(n)}$ , for $ f\in F(n,d)$ . Further the geometric spectral radius of $ A$ was given by $ $ r_g(A)=\max\{\|\lambda\|_2,\;\lambda=(\lambda_1,\cdots,\lambda_d) \in \sigma(A)\}.$Read more