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

For K 1 positive definite symmetric matrices A₁ , ⋯ ,Aₖ of dimension p × p we define the function φ (A₁ , ⋯ ,Aₖ ;n₁ , ⋯ ,nₖ ) = ∏ i = 1ᵏ [ (diag A₁ )]nᵢ /[ (Aᵢ )]nᵢ, where nᵢ are positive constants, as a measure of simultaneous deviation of A₁ , ⋯ ,Aₖ from diagonality. We give an iterative algorithm, called the FG-algorithm, to find an orthogonal p × p-matrix B such that φ (BT A₁ B, ⋯ ,BT Aₖ B;n₁ , ⋯ ,nₖ ) is minimum. The matrix B is said to transform A₁ , ⋯ ,Aₖ 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 Aᵢ.