Canonical Forms for Families of Anti-commuting Diagonalizable Linear Operators

TL;DR

This paper explores the structure of diagonalizable anti-commuting linear operators, showing they can be decomposed into subspaces where they resemble Clifford algebra representations, unlike commuting families which are simultaneously diagonalizable.

Contribution

It establishes a canonical form decomposition for diagonalizable anti-commuting families, linking them to Clifford algebra representations, extending the understanding beyond commuting operators.

Findings

Anti-commuting diagonalizable families cannot be simultaneously diagonalized.

Such families admit an invariant decomposition into subspaces with Clifford algebra structure.

Canonical forms are explicitly constructed for complex and real representations.

Abstract

It is well known that a commuting family of diagonalizable linear operators on a finite dimensional vector space is simultaneously diagonalizable. In this paper, we consider a family A of anti-commuting (complex) linear operators on a finite dimensional vector space V. We prove that if the family is diagonalizable over the complex numbers, then V has an A-invariant direct sum decomposition into subspaces V_a such that the restriction of the family A to V_a is a representation of a Clifford algebra. Thus unlike the families of commuting diagonalizable operators, diagonalizable anti-commuting families cannot be simultaneously digonalized, but on each subspace, they can be put simultaneously to (non-unique) canonical forms. The construction of canonical forms for complex representations is straightforward, while for the real representations it follows from the results of [Bilge A.H., \c{S}. Ko\c{c}ak, S. U\u{g}uz, Canonical Bases for real representations of Clifford algebras, Linear Algebra and its Applications 419 (2006) 417-439. 3].