Abstract

This paper considers the extreme (typically the largest or smallest) singular values of a matrix valued function. A max characterization, using the Frobenius inner product, of the sum of the largest singular values is given. This is obtained by giving a lower bound on the sum of the singular values of a matrix, and necessary and sufficient conditions for attaining this lower bound. The sum f of the largest singular values of a matrix is a convex function of the elements of the matrix, while the smallest singular value is a difference of convex functions. For smooth matrix valued functions these results imply that f is a regular locally Lipschitz function, and a formula for the Clarke subdifferential is given. For a Gâteaux-differentiable matrix-valued function f is a semiregular functions, while the smallest singular value is the negative of a semiregular functions. This enables us to derive concise characterizations of the generalized gradient of functions related to the extreme singular values and the condition number of a matrix.

Author information

Contact details are reproduced from the original publication and may be historical.

Liqun Qi

School of Mathematics, The University of New South Wales, Sydney, N.S.W. 2052, Australia.

Rob S. Womersley

School of Mathematics, The University of New South Wales, Sydney, N.S.W. 2052, Australia.

Keywords

  • singular value
  • subdifferential
  • regularity

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Suggested citation

L. Qi, R. S. Womersley. “On Extreme Singular Values of Matrix Valued Functions.” Journal of Convex Analysis 3 (1996), No. 1, 153–166. https://doi.org/10.68381/jca03011

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Published by Heldermann Verlag, 1996. Rights now held by Banach Press.