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Some Spectral Properties of Polar Decompositions

Published online by Cambridge University Press:  20 November 2018

Bryan E. Cain
Bryan E. Cain*
Affiliation:
Iowa State University, Ames, Iowa
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The results in this paper respond to two rather natural questions about a polar decomposition A = UP, where U is a unitary matrix and P is positive semidefinite. Let λ1, …, λn be the eigenvalues of A. The questions are:

  • (A) When will |λ1|, …, |λn| be the eigenvalues of P?

  • (B) When will λ1/|λ1|, …, λn/|λn| be the eigenvalues of U?

The complete answer to (A) is “if and only if U and P commute.” In an important special case the answer to (B) is “if and only if U2 and P commute.“

Since these matters are best couched in terms of two different inertias, we begin with a unifying definition of inertia which views all inertias from a single perspective.

For each square complex matrix A and each complex number z let m(A, z) denote the multiplicity of z as a root of the characteristic polynomial

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 6 , 01 December 1984 , pp. 973 - 985
Copyright
Copyright © Canadian Mathematical Society 1984

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