Convex Analysis in Spectral Decomposition Systems

Abstract: This work is concerned with convex analysis of so-called spectral functions of matrices that only depend on eigenvalues of the matrix. An abstract framework of spectral decomposition systems is proposed that covers a wide range of previously studied settings, including eigenvalue decomposition of Hermitian matrices and singular value decomposition of rectangular matrices and allows deriving new results in more general settings such as Euclidean Jordan algebras. The main results characterize convexity, lower semicontinuity, Fenchel conjugates, convex subdifferentials, and Bregman proximity operators of spectral functions in terms of the reduced functions. As a byproduct, a generalization of the Ky Fan majorization theorem is obtained.