On the inverse of certain patterned sums of matrices with Kronecker product structures

Abstract

In this article, we derive explicit expressions for the entries of the inverse of a patterned matrix that is a sum of Kronecker products. This matrix keeps the Kronecker structure under matrix inversion, and it is used, for example, in statistics, in particular in the linear mixed model analysis. The obtained results present new and extended existing algorithms for the inversion of the considered patterned matrices. We also obtain a closed-form inverse in terms of block matrices.

Keywords:

• matrix inverse
• Kronecker product
• sum of matrices
• block matrix
• covariance matrix

AMS Subject Classifications::

• 15A09
• 15A69
• 62J10

Acknowledgement

The research was supported by the Estonian Research Foundation Grant GMTMS6702.