Derivatives of spectral functions

• 成果类型: Article
• 署名作者: Lewis, AS
• 刊物名称: MATHEMATICS OF OPERATIONS RESEARCH
• ISSN/ISSBN: 0364-765X
• DOI: 10.1287/moor.21.3.576
• 发表日期: 1996
• 页码: 576-588
• 关键词: Matrices; eigenvalues

摘要

A spectral function of a Hermitian matrix X is a function which depends only on the eigenvalues of X, lambda(1)(X) greater than or equal to lambda(2)(X) greater than or equal to ... greater than or equal to lambda(n)(X), and hence may be written f(lambda(2)(X), lambda(2)(X),..., lambda(n)(X)) for some symmetric function f. Such functions appear in a wide variety of matrix optimization problems. We give a simple proof that this spectral function is differentiable at X if and only if the function f is differentiable at the vector lambda(X), and we give a concise formula for the derivative. We then apply this formula to deduce an analogous expression for the Clarke generalized gradient of the spectral function. A similar result holds for real symmetric matrices.