bv_cvxbook_extra_exercises

# It can be shown that every doubly stochastic matrix

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ) 2.21 Symmetric convex functions of eigenvalues. A function f : Rn → R is said to be symmetric if it is invariant with respect to a permutation of its arguments, i.e., f (x) = f (P x) for any permutation matrix P . An example of a symmetric function is f (x) = log( n=1 exp xk ). k In this problem we show that if f : Rn → R is convex and symmetric, then the function g : Sn → R deﬁned as g (X ) = f (λ(X )) is convex, where λ(X ) = (λ1 (X ), λ2 (x), . . . , λn (X )) is the vector of eigenvalues of X . This implies, for example, that the function n eλ k (X ) g (X ) = log tr eX = log k=1 is convex on Sn . (a) A square matrix S is doubly stochastic if its elements are nonnegative and all row sums and column sums are equal to one. It can be shown that every doubly stochastic matrix is a convex combination of permutation matrices. Show that if f is convex and symmetric and S is doubly stochastic, then f (Sx) ≤ f (x). (b) Let Y = Q diag(λ)QT be an eigenvalue decomposition of Y ∈ Sn with Q orthogonal. Show that the n × n matrix S with elements Sij = Q2 is do...
View Full Document

Ask a homework question - tutors are online