Joint Approximate Diagonalization of Positive Deenite Hermitian Matrices

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

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.

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...

Matrices with Positive Definite Hermitian Part : Inequalities and Linear

The Hermitian and skew-Hermitian parts of a square matrix A are deened by H(A) (A + A)=2 and S(A) (A ? A)=2: We show that the function f(A) = (H(A ?1)) ?1 is convex with respect to the Loewner partial order on the cone of matrices with positive deenite Hermitian part. That is, for any matrices A and B with positive deenite Hermitian part ff(A) + f(B)g=2 ? f(fA + Bg=2) is positive semideenite: U...