Canonical Forms for Families of Anti-commuting Diagonalizable Linear Operators

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].