Canonical forms for families of anti-commuting diagonalizable linear operators
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Date
2012
Authors
Bilge, Ayşe Hümeyra
Journal Title
Journal ISSN
Volume Title
Publisher
Elsevier Science Inc
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.
Description
Keywords
Anti-commuting linear operators, Representations of Clifford algebras
Turkish CoHE Thesis Center URL
Citation
0
WoS Q
Q2
Scopus Q
Q1
Source
Volume
436
Issue
1
Start Page
79
End Page
85