Convexity in Spaces of Matrices

Abstract

In this chapter we investigate three subjects concerning the convexity of functions defined on a space of matrices (or just on a convex subset of it). The first one is devoted to the convex spectral functions, that is, to the convex functions $F:{\text {Sym}}(n,\mathbb {R})\rightarrow \mathbb {R}$ whose values F(A) depend only on the spectrum of A. The main result concerns their description as superpositions $f\circ \Lambda$ between convex functions $f:\mathbb {R}^{n}\rightarrow \mathbb {R}$ invariant under permutations, and the eigenvalues map $\Lambda$.