Diagonality measures of Hermitian positive-definite matrices with application to the approximate joint diagonalization problem

Diagonality Measures of Hermitian Positive-Definite Matrices with Application to the Approximate Joint Diagonalization Problem

In this paper, we introduce properly-invariant diagonality measures of Hermitian positive-definite matrices. These diagonality measures are defined as distances or divergences between a given positive-definite matrix and its diagonal part. We then give closed-form expressions of these diagonality measures and discuss their invariance properties. The diagonality measure based on the log-determin...

Joint Approximate Diagonalization of Positive Definite Hermitian Matrices

Joint Approximate Diagonalization of Positive Deenite Hermitian Matrices

This paper provides an iterative algorithm to jointly approximately di-agonalize K Hermitian positive deenite matrices ? 1 , : : : , ? K. Speciically it calculates the matrix B which minimizes the criterion P K k=1 n k log det diag(BC k B) ? log det(BC k B)], n k being positive numbers, which is a measure of the deviation from diagonality of the matrices BC k B. The convergence of the algorithm...

Newton Method for Joint Approximate Diagonalization of Positive Definite Hermitian Matrices

In this paper we present a Newton method to jointly approximately diagonalize a set of positive definite Hermitian matrices. To this end, we derive the local gradient and Hessian of the underlying cost function in closed form. The algorithm is derived for the complex case and can also update a non-square diagonalization matrix. We analyze the cost function at the critical points and show its re...

Approximate Joint Diagonalization and Geometric Mean of Symmetric Positive Definite Matrices

We explore the connection between two problems that have arisen independently in the signal processing and related fields: the estimation of the geometric mean of a set of symmetric positive definite (SPD) matrices and their approximate joint diagonalization (AJD). Today there is a considerable interest in estimating the geometric mean of a SPD matrix set in the manifold of SPD matrices endowed...