# On the Phases of a Complex Matrix

## TL;DR

This paper introduces the concept of matrix phases for complex sectorial matrices, explores their properties, and applies these insights to stability analysis and matrix completion problems.

## Contribution

It defines matrix phases via canonical angles, studies their properties, and applies these to feedback stability and matrix completion problems.

## Key findings

- Established a majorization relation for eigenvalue phases of matrix products.
- Analyzed the rank of I + AB using phase information.
- Discussed phases of Kronecker and Hadamard products.

## Abstract

In this paper, we define the phases of a complex sectorial matrix to be its canonical angles, which are uniquely determined from a sectorial decomposition of the matrix. Various properties of matrix phases are studied, including those of compressions, Schur complements, matrix products, and sums. In particular, by exploiting a notion known as the compound numerical range, we establish a majorization relation between the phases of the eigenvalues of $AB$ and the phases of $A$ and $B$. This is then applied to investigate the rank of $I + AB$ with phase information of $A$ and $B$, which plays a crucial role in feedback stability analysis. A pair of problems: banded sectorial matrix completion and decomposition is studied. The phases of the Kronecker and Hadamard products are also discussed.