A Generalization of Kristof's Theorem on the Trace of Certain Matrix Products

Kristof has derived a theorem on the maximum and minimum of the trace of matrix products of the form [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document} X₁ Γ ₁ X₂ Γ ₂ ⋯ Xₙ Γ ₙ {document} where the matrices [12pt]{minimal} {amsmath} {wasysym} {amsfonts} {amssymb} {amsbsy} {mathrsfs} {upgreek} {}{-69pt} {document} Γ ᵢ {document} are diagonal and fixed and the X i vary unrestrictedly and independently over the set of orthonormal matrices. The theorem is a useful tool in deriving maxima and minima of matrix trace functions subject to orthogonality constraints. The present paper contains a generalization of Kristof's theorem to the case where the X i are merely required to be submatrices of orthonormal matrices and to have a specified maximum rank. The generalized theorem contains the Schwarz inequality as a special case. Various examples from the psychometric literature, illustrating the practical use of the generalized theorem, are discussed.