A Simultaneous Cholesky Decomposition of Two Positive Definite Matrices

다국어 초록 (Multilingual Abstract)

In this paper we consider the simultaneous diagonalization of two positive definite matrices. A necessary and sufficient condition is found under which two positive definite covariance matrices are simultaneously diagonalized in view of a Cholesky decomposition. This result can be applied to estimating the eigenvalues and eigenvectors of for two positive definite covariance matrices ,. under multivariate normality. The result is extended to more than two matrices. The result is also applied to computing the conditional expectations contained in some decompositions of the Mahalanobis distance that help us to explain some reasons for the outlyingness of multivariate observations in multivariate analysis. Since the Cholesky decomposition of covariance matrix is useful in expressing the conditional expectation of one variable when the remaining variables are fixed under multivariate normality. An example is given for an illustration.

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