Constrained optimization of the overdetermined Zimmermann compressor for nonorthogonal joint matrix diagonalization

Abstract

The classical $2 \times 2$ Zimmermann matrix known from nonorthogonal Jacobi algorithms for symmetric definite matrix pairs can be used as an elementary compressor in the overdetermined problem of joint matrix diagonalization of more than two matrices. For this purpose, a constrained optimization procedure for adapting the overdetermined Zimmermann compressor to offdiagonal error minimization is established. It turns out that in the case of the Zimmermann matrix, the constrained optimization approach exhibits a key feature in terms of a $3 \times 3$ rank-deficient constraint matrix. This rank drop simplifies the complexity of Zimmermann matrix design relative to other possible types of compressor matrices in this application. A minimum complexity nonorthogonal Jacobi algorithm for joint matrix diagonalization is hence obtained on the basis of the overdetermined Zimmermann compressor. We show diagonalization results obtained with this algorithm for extreme cases like the joint diagonalization of only $3$ highly-correlated matrices of dimension $500 \times 500$. A “null space” variant of the classical Zimmermann algorithm for matrix pairs is obtained as a by-product from the overdetermined joint matrix diagonalizer.