  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].
