Quaternionic matrices: Unitary similarity, simultaneous triangularization and some trace identities

TL;DR

This paper develops invariants for classifying 2x2 quaternionic matrices under unitary similarity, extends triangularization characterizations to quaternionic matrices, and explores simultaneous triangularization problems.

Contribution

It introduces minimal unitary trace invariants for 2x2 quaternionic matrices and extends classical triangularization results to the quaternionic setting.

Findings

Six minimal unitary trace invariants for 2x2 quaternionic matrices

Quaternionic versions of triangularization characterizations

Analysis of the semi-algebraic set of simultaneously triangularizable pairs

Abstract

We construct six unitary trace invariants for 2 by 2 quaternionic matrices which separate the unitary similarity classes of such matrices, and show that this set is minimal. We prove two quaternionic versions of a well known characterization of triangularizable subalgebras of matrix algebras over an algebraically closed field. Finally we consider the problem of describing the semi-algebraic set of pairs (X,Y) of quaternionic n by n matrices which are simultaneously triangularizable. Even the case n=2, which we analyze in more detail, remains unsolved.