Determinant of a Sum of Certain Kronecker Products

TL;DR

This paper derives a formula for computing the determinant of a sum of Kronecker products where one factor is square and the other is a rank-one matrix formed from vectors, simplifying calculations in related matrix problems.

Contribution

It provides a novel explicit formula for the determinant of a sum of specific Kronecker products involving rank-one matrices, expanding computational tools in matrix analysis.

Findings

Derived a closed-form determinant expression for the sum of Kronecker products.

Applicable to matrices with rank-one components, simplifying determinant calculations.

Enhances understanding of structure in sums of Kronecker products.

Abstract

We compute the determinant of $\sum_{n=1}^{N} \vec{A}^{(n)} \otimes \vec{B}^{(n)}$, where $\vec{A}^{(n)}$ is square and ${\vec{B}^{(n)}=\vec{x}^{(n)}{\vec{y}^{(n)}}^T}$ where $\vec{x}^{(n)}$ and $\vec{y}^{(n)}$ have length $N$.