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A non-commutative spectral theorem

Authors
Journal
Linear Algebra and its Applications
0024-3795
Publisher
Elsevier
Publication Date
Volume
20
Issue
2
Identifiers
DOI: 10.1016/0024-3795(78)90043-5
Disciplines
  • Mathematics

Abstract

Abstract Let M p denote the full algebra of pX p matrices, and let M ( k) p denote the algebra of ( pk)X( pk) matrices of the form diag( B,…, B), where B ∈ M p and there are k blocks. We show that if is an algebra of nX n matrices which is generated by a set of normal matrices, there is a unitary matrix U such that for each A ∈ we have U ∗AU= diag(B 1,…,B m,0) where B i ∈ M ( k i ) Pi ; 0 is a zero matrix of some order, say r; and n= r+ Σ n i=1 p i k i . The result is applied to several algebras which satisfy polynomial identities.

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