Journal
LINEAR ALGEBRA AND ITS APPLICATIONS
Volume 432, Issue 5, Pages 1234-1257Publisher
ELSEVIER SCIENCE INC
DOI: 10.1016/j.laa.2009.10.034
Keywords
Eigenvalues; Singular values; Rayleigh quotient; Orthogonal quotient matrices; The orthogonal quotients equality; Eckart-Young theorem; Fan's extremum principles
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In this paper we explore the extremum properties of orthogonal quotients matrices. The orthogonal quotients equality that we prove expresses the Frobenius norm of a difference between two matrices as a difference between the norms of two matrices. This turns the Eckart-Young minimum norm problem into an equivalent maximum norm problem. The symmetric version of this equality involves traces of matrices, and adds new insight into Fan's extremum problems. A comparison of the two cases reveals a remarkable similarity between the Eckart-Young theorem and Fan's maximum principle. Returning to orthogonal quotients matrices we derive rectangular extensions of Fan's extremum principles, which consider maximizing (or minimizing) sums of powers of singular values. (C) 2009 Elsevier Inc. All rights reserved.
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