Canonical forms for families of anti-commuting diagonalizable linear operators

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={Aa}, Aa:V→V, a=1,…,N of anti-commuting (complex) linear operators on a finite dimensional vector space. We prove that if the family is diagonalizable over the complex numbers, then V has an A-invariant direct sum decomposition into subspaces Vα such that the restriction of the family A to Vα 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 (nonunique) canonical forms. The construction of canonical forms for complex representations is straightforward, while for the real representations it follows from the results of [A.H. Bilge, Ş. Koçak, S. Uğuz, Canonical bases for real representations of Clifford algebras, Linear Algebra Appl. 419 (2006) 417–439].