| Abstract | We are all familiar with the spectral decomposition or eigenvalue-eigenvector decomposition of a matrix. This paper addresses cases where instead of using a single eigenvalue we may use a square non-diagonal matrix, which we will call eigenmatrix, which goes along with the concomitant set or matrix of block-eigenvectors in a number that has to match the dimension of the eigenmatrix. The usefulness of such construct or decomposition in statistical studies in multivariate analysis, more precisely, in the derivation of likelihood ratio statistics for tests of elaborate structures of covariance matrices, their moments and their distributions, is shown, and an application is made to the derivation and study of the likelihood ratio statistic to test block-circularity of covariance matrices. Through the use of such construct or decomposition, also near-exact distributions are easily obtained for this statistic and the relation of the test with other tests is easily derived. |
