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

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...

Joint Approximate Diagonalization of Positive Definite Hermitian Matrices

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...

Matrix commutators: their asymptotic metric properties and relation to approximate joint diagonalization

We analyze the properties of the norm of the commutator of two Hermitian matrices, showing that asymptotically it behaves like a metric, and establish its relation to joint approximate diagonalization of matrices, showing that almost-commuting matrices are almost jointly diagonalizable, and vice versa. We show an application of our results in the field of 3D shape analysis.

Stair Matrices and Their Generalizations with Applications to Iterative Methods Ii: Iteration Arithmetic and Preconditionings

Iteration arithmetic is formally introduced based on iteration multiplication and αaddition which is a special multisplitting. This part focuses on construction of convergent splittings and approximate inverses for Hermitian positive definite matrices by applying stair matrices, their generalizations and iteration arithmetic. Analysis of the splittings and the approximate inverses is also prese...