A generalization of Kristof’s theorem on the trace of certain matrix products. (English) Zbl 0536.62093

Summary: W. Kristof [J. Math. Psychol. 7, 515-530 (1970; Zbl 0205.492)] has derived a theorem on the maximum and minimum of the trace of matrix products of the form \(X_ 1{\hat \Gamma}_ 1X_ 2{\hat \Gamma}_ 2\cdot \cdot \cdot X_ n{\hat \Gamma}_ n\) where the matrices \({\hat \Gamma}{}_ i\) are diagonal and fixed and the \(X_ i\) vary unrestricted and independently over the set of orthogonal 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.


62P15 Applications of statistics to psychology
15A99 Basic linear algebra
15A45 Miscellaneous inequalities involving matrices


Zbl 0205.492
Full Text: DOI


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