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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.

MSC:

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

Citations:

Zbl 0205.492
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References:

[1] Bock, R. D.Multivariate statistical methods in behavioral research. New-York: McGraw-Hill, 1975. · Zbl 0398.62086
[2] Green, B. F. Best linear composites with a specified structure.Psychometrika, 1969,34, 301–318. · Zbl 0177.46401
[3] Kristof, W. A theorem on the trace of certain matrix products and some applications.Journal of Mathematical Psychology, 1970,7, 515–530. · Zbl 0205.49202
[4] Levin, J. Applications of a theorem on the traces of certain matrix products.Multivariate Behavioral Research, 1979,1, 109–113.
[5] Von Neumann, J. Some matrix-inequalities and metrization of matrix-space.Tomsk University Review, 1937, 1, 286–300. Reprinted in A. H. Taub (Ed.),John von Neumann collected works (Vol. IV). New-York: Pergamon, 1962. · JFM 63.0037.03
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