Projectively and weakly simultaneously diagonalizable matrices and their applications

TL;DR

This paper introduces new weaker variants of simultaneously diagonalizable matrices, explores their properties, and demonstrates their applications in quadratic programming and independent component analysis.

Contribution

The paper proposes several novel weaker forms of SD matrices, providing conditions and relationships, expanding the applicability of diagonalization techniques.

Findings

New variants of SD matrices introduced

Conditions and relationships between variants established

Applications demonstrated in QCQP and ICA

Abstract

Characterizing simultaneously diagonalizable (SD) matrices has been receiving considerable attention in the recent decades due to its wide applications and its role in matrix analysis. However, the notion of SD matrices is arguably still restrictive for wider applications. In this paper, we consider two error measures related to the simultaneous diagonalization of matrices, and propose several new variants of SD thereof; in particular, TWSD, TWSD-B, T_{m,n}-SD (SDO), DWSD and D_{m,n}-SD (SDO). Those are all weaker forms of SD. We derive various sufficient and/or necessary conditions of them under different assumptions, and show the relationships between these new notions. Finally, we discuss the applications of these new notions in, e.g., quadratically constrained quadratic programming (QCQP) and independent component analysis (ICA).