Bounds on eigenvalues of product of matrices

Question

I have 4 matrices $\mathbf{V}_1$, $\mathbf{V}_2$, $\mathbf{V}_3$, $\mathbf{V}_4$, all with eigenvalues $0 < |\lambda^i_n| <1$, where $\lambda^i_n$ is the $n$th eigenvalue of the $i$th matrix, can I infer that $\mathbf{V}=\mathbf{V}_4 \mathbf{V}_3 \mathbf{V}_2 \mathbf{V}_1 $ also has eigenvalues with modulus less than 1?

Answer 1

I think you can use Gelfand's corollaries.

Let $\mathbf V$ be a matrix in $\mathbf{C}^{n\times n}$, we define the spectral radius $$\rho(\mathbf V) = \max_i |\lambda_i|$$ where $\lambda_i$ are the eigenvalues of $\mathbf V$. By assumption, we have that $\rho(\mathbf V_j)<1$ for $j=1,\ldots,4$.

By Gelfand's corollaries $$\rho(\mathbf V) = \rho(\mathbf V_1 \mathbf V_2 \mathbf V_3 \mathbf V_4) \leq \rho(\mathbf V_1)\rho( \mathbf V_2 )\rho(\mathbf V_3)\rho( \mathbf V_4) < 1$$ as it is the product of four numbers that are less than 1.