# Eigenvalues of a product of matrices with specific structures

I'm working on a multichannel feedback system with open-loop frequency response $$\mathbf{A}(\omega) = \mathbf{B}(\omega)(\mathbf{C}(\omega)-\mathbf{D}(\omega))$$. The time-domain convolution matrices corresponding to the different frequency responses have the following characteristics: $$\mathbf{B}$$ is a hollow matrix (i.e. a matrix with all zeros along the main diagonal), $$\mathbf{C}$$ is a dense matrix and $$\mathbf{D}$$ is a band matrix (not completely sure it can be qualified as such, see picture). All the matrices have a block Toeplitz structure are full rank. An example is the following, showing the three matrices and the resulting time-domain resulting matrix $$\mathbf{A} = \mathbf{B}(\mathbf{C}-\mathbf{D})$$:

I was wondering if, starting from such a structure, anything can be inferred about the eigenvalues of the matrix $$\mathbf{A}(\omega)$$, whose behaviour I want to study as they have an impact on the stability of the system. Specifically, I'd be interested in knowing if their moduli can be inferred to be positive.

Or do you think the information on the structure is irrelevant to be able to get the information I need on the eigenvalues?

• I don't know if I can answer the question (I probably can't), but I'm just curious: are the matrices symmetric as well, or just Toeplitz? – Tendero Oct 11 '18 at 14:59
• @Tendero: Just Toeplitz. As the updated image shows, I forgot to mention there's a lower triangular structure that could be accounted for, though. – gbernardi Oct 12 '18 at 6:02
• Can you talk a little bit more about the matrices? Do these matrices describe 1 system? So, does your input vector really have a few tens of thousands of data points? – A_A Oct 12 '18 at 8:32
• All the matrices represent MIMO acoustic LTI systems. So each block is the Toeplitz matrix associated with an acoustic RIR (i.e. an FIR filter). The lengths of the filters might change in reality. Here I simply kept roughly half a second (4096 samples at 8kHz) per FIR. – gbernardi Oct 12 '18 at 12:37
• Do you have any information about the eigenvalues of $B$, $C$ and $D$? Or do you just wanna figure out if, with no information about them, one can make some inference about $A$'s eigenvalues due to the structure of the matrices only (and not their spectrum)? – Tendero Oct 12 '18 at 14:46