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An extension of the Minkowski determinant theorem

Published online by Cambridge University Press:  20 January 2009

Marvin Marcus  and
William R. Gordon
Marvin Marcus
Affiliation:
University of California, Santa Barbara
William R. Gordon
Affiliation:
University of Victoria, Victoria B. C.
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Minkowski proved the following (for a proof see (4)): if A and B are n × n positive semi-definite hermitian matrices then

It is known (4) that if both A and B are non-singular, then the equality holds in (1) if and only if B = cA where c is a positive number.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 17 , Issue 4 , December 1971 , pp. 321 - 324
Copyright
Copyright © Edinburgh Mathematical Society 1971

References

REFERENCES

(1) Birkhoff, G., Tres observaciones sobre el algebra lineal, Univ. Nac. Tucuman Rev., Ser. A 5 (1946), 147–150.Google Scholar
(2) Marcus, M. and Minc, H., A Survey of Matrix Theory and Matrix Inequalities (Prindle, Weber and Schmidt, Boston, Mass., 1964).Google Scholar
(3) Gordon, W. R. and Marcus, M., An analysis of equality in certain matrix inequalities I, Pacific J. Math. 34 (1970), 407–413.Google Scholar
(4) Mirsky, L., An Introduction to Linear Algebra (Oxford 1955), 419.Google Scholar