On Extreme Singular Values of Matrix Valued Functions

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 of a matrix is a convex function of the 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 characterizations of the generalized gradient of functions related to the extreme singular values and the condition number of a matrix.