An Algorithm for Simultaneous Orthogonal Transformation of Several Positive Definite Symmetric Matrices to Nearly Diagonal Form

Abstract

For $K \geqq 1$ positive definite symmetric matrices $A_1 , \cdots ,A_k $ of dimension $p \times p$ we define the function $\phi (A_1 , \cdots ,A_k ;n_1 , \cdots ,n_k ) = \prod _{i = 1}^k [\det (\operatorname{diag} A_1 )]^{n_i } /[\det (A_i )]^{n_i } $, where $n_i $ are positive constants, as a measure of simultaneous deviation of $A_1 , \cdots ,A_k $ from diagonality. We give an iterative algorithm, called the FG-algorithm, to find an orthogonal $p \times p$-matrix B such that $\phi (B^T A_1 B, \cdots ,B^T A_k B;n_1 , \cdots ,n_k )$ is minimum. The matrix B is said to transform $A_1 , \cdots ,A_k $ simultaneously to nearly diagonal form. Conditions for the uniqueness of the solution are given. The FG-algorithm can be used to find the maximum likelihood estimates of common principal components in k groups (Flury (1984)). For $k = 1$, the FG-algorithm computes the characteristic vectors of the single positive definite symmetric matrix $A_i $.