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.
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References
Aitken, A.C.: Determinants and Matrices. Oliver & Boyd, Edinburgh-London (1952)
Gröbner, W.: Matrizenrechnung. Bibliographisches Institut, Mannheim (1965)
Lancaster, P.: Theory of Matrices. Academic, New York (1969)
Marshall, A.W., Olkin, I.: Inequalities: Theory of Majorization and its Applications. Academic, New York (1979)
Meyer, C.D.: Matrix Analysis and Applied Linear Algebra. SIAM, Philadelphia, USA (2000)
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© 2009 Physica-Verlag Heidelberg
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Werner, H.J., Olkin, I. (2009). On Permutations of Matrix Products. In: Schipp, B., Kräer, W. (eds) Statistical Inference, Econometric Analysis and Matrix Algebra. Physica-Verlag HD. https://doi.org/10.1007/978-3-7908-2121-5_25
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DOI: https://doi.org/10.1007/978-3-7908-2121-5_25
Publisher Name: Physica-Verlag HD
Print ISBN: 978-3-7908-2120-8
Online ISBN: 978-3-7908-2121-5
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