Matrix diagonalization is an important component of many aspects of computational science. There are a variety of algorithms to accomplish this task. Jacobi’s algorithm is a good choice for parallel environments. Jacobi’s algorithm consists of a series of matrix plane rotations, the ordering of which can dramatically affect performance. We show a new ordering which cuts the number of necessary ...

In this paper we study the effect of block diagonalization of a nearly diagonal matrix by iterating the related Riccati equations. We show that the iteration is fast, if a matrix is diagonally dominant or scaled diagonally dominant and the block partition follows an appropriately defined spectral gap. We also show that both kinds of diagonal dominance are not destroyed after the block diagonali...

We present a new scale-invariant cost function for non-orthogonal joint-diagonalization of a set of symmetric matrices with application to Independent Component Analysis (ICA). We derive two gradient minimization schemes to minimize this cost function. We also consider their performance in the context of an ICA algorithm based on non-orthogonal joint diagonalization.

Lemma 1.1. Let V be a finite-dimensional vector space over a field F. Let β, β′ be two bases for V . Let T : V → V be a linear transformation. Define Q := [IV ] ′ β . Then [T ] β β and [T ] ′ β′ satisfy the following relation [T ] ′ β′ = Q[T ] β βQ −1. Theorem 1.2. Let A be an n× n matrix. Then A is invertible if and only if det(A) 6= 0. Exercise 1.3. Let A be an n×n matrix with entries Aij, i,...