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A THEORETICAL STUDY OF THE OBJECTIVE-FUNCTIONAL FOR JOINT EIGEN-DECOMPOSITION OF MATRICES

Troedsson, Erik LU ; Falkowski, Daniel ; Lidgren, Carl Fredrik ; Wendt, Herwig and Carlsson, Marcus LU (2025) In SIAM Journal on Matrix Analysis and Applications 46(3). p.2061-2079
Abstract

The problem of approximate joint eigen-decomposition of a collection of matrices arises in a number of diverse engineering and signal processing problems. This problem is usually cast as an optimization problem, and it is the main goal of this publication to provide a theoretical study of the corresponding objective-functional. As our main result, we prove that this functional tends to infinity in the vicinity of rank-deficient matrices with probability one, thereby proving that the optimization problem is well posed. Second, we provide unified expressions for its higher order derivatives in multilinear form, and explicit expressions for the gradient and the Hessian of the functional in standard form, thereby allowing for new improved... (More)

The problem of approximate joint eigen-decomposition of a collection of matrices arises in a number of diverse engineering and signal processing problems. This problem is usually cast as an optimization problem, and it is the main goal of this publication to provide a theoretical study of the corresponding objective-functional. As our main result, we prove that this functional tends to infinity in the vicinity of rank-deficient matrices with probability one, thereby proving that the optimization problem is well posed. Second, we provide unified expressions for its higher order derivatives in multilinear form, and explicit expressions for the gradient and the Hessian of the functional in standard form, thereby allowing for new improved numerical schemes for the solution of the joint eigen-decomposition problem. A special section is devoted to the important case of self-adjoint matrices.

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author
; ; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Fréchet derivative, joint eigenvalue decomposition, matrix diagonalization, nonlinear functional, simultaneous diagonalization, well-posedness
in
SIAM Journal on Matrix Analysis and Applications
volume
46
issue
3
pages
19 pages
publisher
Society for Industrial and Applied Mathematics
external identifiers
  • scopus:105015649759
ISSN
0895-4798
DOI
10.1137/24M171351X
language
English
LU publication?
yes
id
022f8ee2-c091-4b76-993c-fa2f6199d9fd
date added to LUP
2025-11-12 14:39:41
date last changed
2025-11-12 14:39:41
@article{022f8ee2-c091-4b76-993c-fa2f6199d9fd,
  abstract     = {{<p>The problem of approximate joint eigen-decomposition of a collection of matrices arises in a number of diverse engineering and signal processing problems. This problem is usually cast as an optimization problem, and it is the main goal of this publication to provide a theoretical study of the corresponding objective-functional. As our main result, we prove that this functional tends to infinity in the vicinity of rank-deficient matrices with probability one, thereby proving that the optimization problem is well posed. Second, we provide unified expressions for its higher order derivatives in multilinear form, and explicit expressions for the gradient and the Hessian of the functional in standard form, thereby allowing for new improved numerical schemes for the solution of the joint eigen-decomposition problem. A special section is devoted to the important case of self-adjoint matrices.</p>}},
  author       = {{Troedsson, Erik and Falkowski, Daniel and Lidgren, Carl Fredrik and Wendt, Herwig and Carlsson, Marcus}},
  issn         = {{0895-4798}},
  keywords     = {{Fréchet derivative; joint eigenvalue decomposition; matrix diagonalization; nonlinear functional; simultaneous diagonalization; well-posedness}},
  language     = {{eng}},
  number       = {{3}},
  pages        = {{2061--2079}},
  publisher    = {{Society for Industrial and Applied Mathematics}},
  series       = {{SIAM Journal on Matrix Analysis and Applications}},
  title        = {{A THEORETICAL STUDY OF THE OBJECTIVE-FUNCTIONAL FOR JOINT EIGEN-DECOMPOSITION OF MATRICES}},
  url          = {{http://dx.doi.org/10.1137/24M171351X}},
  doi          = {{10.1137/24M171351X}},
  volume       = {{46}},
  year         = {{2025}},
}