Partial determinants of Kronecker products

TL;DR

This paper investigates the properties of partial determinants of Kronecker products, characterizing conditions for determinant equality and introducing a determinant-root operation with multiplicative properties.

Contribution

It provides a characterization of when the partial determinant of a Kronecker product equals the determinant of the original matrix and introduces a new determinant-root operation.

Findings

Conditions for $ ext{det}_2(A)$ to satisfy $ ext{det}( ext{det}_2(A)) = ext{det}(A)$

A characterization of matrices forming a multiplicative monoid under $ ext{det}_2$

Definition of a determinant-root operation $ ext{Det}$ with specific compositional properties.

Abstract

Let $\det_2(A)$ be the block-wise determinant (partial determinant). We consider the condition for completing the determinant $\det(\det_2(A)) = \det(A),$ and characterize the case for an arbitrary Kronecker product $A$ of matrices over an arbitrary field. Further insisting that $\det_2(AB)=\det_2(A)\det_2(B)$, for Kronecker products $A$ and $B$, yields a multiplicative monoid of matrices. This leads to a determinant-root operation $\text{Det}$ which satisfies $\text{Det}(\text{Det}_2(A)) = \text{Det}(A)$ when $A$ is a Kronecker product of matrices for which $\text{Det}$ is defined.