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 = {A(a)}, A(a) : 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 V(alpha) such that the restriction of the family A to V(alpha) 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 [A.H. Bilge, S. Kocak, S. Uguz, Canonical bases for real representations of Clifford algebras, Linear Algebra Appl. 419 (2006) 417-439]. (C) 2011 Elsevier Inc. All rights reserved

On families of anticommuting matrices

“…Proof. As in the proof of Theorem 13, we can assume that the matrices have the form (6). Note that e ′ 2 , e ′′ 2 are invertible and e ′ 2 e ′′ 3 , .…”

“…where e ′ i , e ′′ i ∈ C n/2×n/2 are invertible. This is because, by Proposition 11, we can write e 1 as in (6) with σ(e ′ 1 ) = {λ}, σ(e ′′ 1 ) = {−λ}, λ = 0. Lemma 10 part (ii) gives that every e j , j > 1 must indeed be of the form required in (6).…”

“…Proof. As in the proof of Theorem 13, we can assume that the matrices have the form (6) If (e ′ 2 e ′′ j ) 2 = 0, the first equality gives e ′ j e ′′ j = 0 and the second e ′′ j e ′ j = 0 (recall that e ′ 2 , e ′′ 2 are invertible). But since e 2 j = e ′ j e ′′ j ⊕ e ′′ j e ′ j , this gives e 2 j = 0 -contrary to the assumption e 2 j = 0.…”

“…Such a problem can be solved under some extra assumptions. In [6], it was shown that an anticommuting family of diagonalisable matrices can be "decomposed" into representations of Clifford algebras. This indeed answer Question 1 if the e i 's are diagonalisable.…”