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 (A₁, , Aₖ ;n₁, , nₖ) = ₈ = 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 (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.