On extremum properties of orthogonal quotients matrices

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 int...

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Bibliographic Details
Published in:Linear algebra and its applications 2010-02, Vol.432 (5), p.1234-1257
Main Author: Dax, Achiya
Format: Article
Language:English
Subjects:
Algebra
Eckart–Young theorem
Eigenvalues
Exact sciences and technology
Fundamental areas of phenomenology (including applications)
Ky Fan’s Extremum principles
Linear and multilinear algebra, matrix theory
Mathematics
Orthogonal quotient matrices
Physics
Rayleigh quotient
Sciences and techniques of general use
Singular values
Solid mechanics
Structural and continuum mechanics
The orthogonal quotients equality
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Summary: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 Ky Fan’s extremum problems. A comparison of the two cases reveals a remarkable similarity between the Eckart–Young theorem and Ky Fan’s maximum principle. Returning to orthogonal quotients matrices we derive “rectangular” extensions of Ky Fan’s extremum principles, which consider maximizing (or minimizing) sums of powers of singular values.
ISSN:0024-3795
1873-1856
DOI:10.1016/j.laa.2009.10.034