Abstract
Kristof has derived a theorem on the maximum and minimum of the trace of matrix products of the form\(X_1 \hat \Gamma _1 X_2 \hat \Gamma _2 \cdots X_n \hat \Gamma _n\) where the matrices\(\hat \Gamma _i\) are diagonal and fixed and theX 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 theX 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.
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The author is obliged to Frits Zegers and Dirk Knol for critically reviewing a previous draft of this paper.
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Ten Berge, J.M.F. A generalization of Kristof's theorem on the trace of certain matrix products. Psychometrika 48, 519–523 (1983). https://doi.org/10.1007/BF02293876
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DOI: https://doi.org/10.1007/BF02293876