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On Permutations of Matrix Products

In: Statistical Inference, Econometric Analysis and Matrix Algebra

Author

Listed:
  • Hans Joachim Werner

    (Universität Bonn, Wirtschaftswissenschaftlicher Fachbereich, Statistische Abteilung)

  • Ingram Olkin

    (Stanford University, Department of Statistics)

Abstract

It is well-known that trace(AB) ≥ 0 for real-symmetric nonnegative definite matrices A and B. However, trace(ABC) can be positive, zero or negative, even when C is real-symmetric nonnegative definite. The genesis of the present investigation is consideration of a product of square matrices A =A 1 A 2 …A n. Permuting the factors of A leads to a different matrix product. We are interested in conditions under which the spectrum remains invariant. The main results are for square matrices over an arbitrary algebraically closed commutative field. The special case of real-symmetric, possibly nonnegative definite, matrices is also considered.

Suggested Citation

  • Hans Joachim Werner & Ingram Olkin, 2009. "On Permutations of Matrix Products," Springer Books, in: Bernhard Schipp & Walter Kräer (ed.), Statistical Inference, Econometric Analysis and Matrix Algebra, pages 359-365, Springer.
    • Handle: RePEc:spr:sprchp:978-3-7908-2121-5_25
    DOI: 10.1007/978-3-7908-2121-5_25
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