On the Convexity of log det (I + K X^{-1})

A simple proof is given for the convexity of log det(I +KX) in the positive definite matrix variable X ≻ 0 with a given positive semidefinite K 0. Convexity of functions of covariance matrices often plays an important role in the analysis of Gaussian channels. For example, suppose Y and Z are independent complex Gaussian nvectors with Y ∼ N(0,K) and Z ∼ N(0,X). Then, I(Y;Y + Z) = log det(I +KX). (1) The following result is well known in the literature. Lemma 1. For a fixed K 0, log det(I +KX) is convex in X ≻ 0, with strict convexity if K ≻ 0. A simple information theoretic proof was given by Diggavi and Cover [4, Lemma II.3] as a corollary to their main saddle-point theorem stating that the worst additive noise is Gaussian. Very recently Mao et al. [6] gave a different, but complicated proof, correcting an incomplete approach taken in Kashyap et al. [5]. The main purpose of this note is to introduce an elegant and simple proof technique based on the theory of spectral functions of Hermitian matrices, which will hopefully benefit other problems in information theory with similar structure. A real-valued function f(X) of Hermitian argumentX ∈ Rn×n is called a spectral function if the value of f(X) depends only on (unordered) eigenvalues of X. If λ(X) ∈ Rn denotes the ordered eigenvalues of X and g is a real-valued symmetric (=permutation invariant) function on Rn, the composite function (g ◦ λ)(X) = g(λ(X)) is a spectral function. Conversly, any spectral function f(X) can be decomposed in this way. It is also easy to see that f(X) is a spectral function if and only if f(X) is unitary invariant, that is, f(X) = f(QXQ) for any unitary Q. Convexity of a spectral function can be checked rather easily; a spectral function f = g ◦ λ is (strictly) convex if and only if the corresponding symmetric function g is (strictly) convex [3]. In other wodrs, a spectral function f(X) is convex for all Hermitian X if and only if f(X) is convex for all real diagonal X. Examples of convex spectral functions include the trace, the largest eigenvalue, and the sum of the k largest eigenvalues, of a Hermitian matrix; and the trace of the inverse of a positive definite matrix (as well as the log determinant of the ∗Email: [email protected] and [email protected]