Zero patterns and unitary similarity

TL;DR

This paper investigates zero patterns in traceless matrices and characterizes which patterns allow every matrix to be unitarily similar to a matrix with zeros in specified positions, extending classical triangularization results.

Contribution

The paper introduces the concept of universal zero patterns in traceless matrices, providing a full characterization for small sizes and constructing infinite families of such patterns.

Findings

Full description of universal patterns for n ≤ 3

Partial results for n = 4

Construction of two infinite families of universal patterns

Abstract

A subspace of the space, L(n), of traceless complex $n\times n$ matrices can be specified by requiring that the entries at some positions $(i,j)$ be zero. The set, $I$, of these positions is a (zero) pattern and the corresponding subspace of L(n) is denoted by $L_I(n)$. A pattern $I$ is universal if every matrix in L(n) is unitarily similar to some matrix in $L_I(n)$. The problem of describing the universal patterns is raised, solved in full for $n\le3$, and partial results obtained for $n=4$. Two infinite families of universal patterns are constructed. They give two analogues of Schur's triangularization theorem.