Essential Mathematics for Convex Optimization

In this chapter, we demonstrate that (a) substituting the vector of eigenvalues of a symmetric n x n matrix into a convex permutation symmetric function of n real variables results in a convex function of the matrix, and (b) that if g is a convex function on the real axis, and G is the set of symmetric matrices of a given size with spectrum in the domain of g, then G is a convex set, and when X is a matrix from G, the trace of the matrix g(X), is a convex function of X; here g(X) is the matrix acting at a spectral subspace of X associated with eigenvalue v as multiplication by g(v); both these facts will be heavily used when speaking about cone-convexity is chapter 21.