Nonsmooth Analysis of Singular Values. Part I: Theory

Abstract The singular values of a rectangular matrix are nonsmooth functions of its entries. In this work we study the nonsmooth analysis of functions of singular values. In particular we give simple formulae for the regular subdifferential, the limiting subdifferential, and the horizon subdifferential, of such functions. Along the way to the main result we give several applications and in particular derive von Neumann’s trace inequality for singular values.

[1]  N. Wiegmann,et al.  Some analogs of the generalized principal axis transformation , 1948 .

[2]  On Pearl's Paper "A Decomposition Theorem for Matrices"* , 1969, Canadian Mathematical Bulletin.

[3]  F. Brickell,et al.  Differentiable manifolds : an introduction , 1970 .

[4]  Peter M. Gibson,et al.  Simultaneous diagonalization of rectangular complex matrices , 1974 .

[5]  W. Greub.  Linear Algebra , 1981 .

[6]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[7]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[8]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[9]  F. Clarke.  Necessary conditions for nonsmooth problems in-optimal control and the calculus of variations , 1991 .

[10]  David J. Goodman,et al.  Personal Communications , 1994, Mobile Communications.

[11]  A. Lewis.  The Convex Analysis of Unitarily Invariant Matrix Functions , 1995 .

[12]  Adrian S. Lewis,et al.  Convex Analysis on the Hermitian Matrices , 1996, SIAM J. Optim..

[13]  A. S. Lewis,et al.  Derivatives of Spectral Functions , 1996, Math. Oper. Res..

[14]  A. S. Lewis,et al.  Group Invariance and Convex Matrix Analysis , 1996, SIAM J. Matrix Anal. Appl..

[15]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[16]  Adrian S. Lewis,et al.  Nonsmooth analysis of eigenvalues , 1999, Math. Program..

[17]  T. Tam.  An extension of a result of Lewis , 1999 .

[18]  Hristo S. Sendov.  Variational Spectral Analysis , 2000 .

[19]  A. Lewis.  Convex analysis on Cartan subspaces , 2000 .

[20]  Hristo S. Sendov,et al.  Nonsmooth Analysis of Singular Values. Part II: Applications , 2005 .