On the Simultaneous Sub-diagonalization and Diagonalization of a Set of Symmetric Matrices

Abstract

This work aims to introduce and discuss the new achievement on matrix theory that leaded to the development of the new unbiased estimator for variance components in mixed linear models (MLM) with an arbitrary number of variance components called Sub-D; in a more detailed way, it introduces and discuss the notion of simultaneous sub-diagonalization of any set of symmetric matrices and then discuss the conditions under which there exists an orthogonal matrix that simultaneous diagonalizes all matrices in that set, using the spectral decomposition. For the particular case when all matrices commutes among themselves the existence of such orthogonal matrix is obvious, but we have no interest in such situation since, in general, the matrices that make up the random part of the MLM do not commute among themselves. The discussion will be firstly conducted over the particular case of a set with two symmetric matrices, and then it will be generalized for the case of a set with an arbitrary number of symmetric matrices. Moreover, besides ensuring the existence of such an orthogonal matrix through an equivalent condition, this work provides the corresponding algorithm.