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 \(\Lambda \) .