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On the Positive Semidefinite Nature of a Certain Matrix Expression

Published online by Cambridge University Press:  20 November 2018

Eugene P. Wigner  and
Mutsuo M. Yanase
Eugene P. Wigner
Affiliation:
Princeton University and Institute for Advanced Study, Princeton
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By "positive definite matrices" or, briefly, definite matrices, we mean in this note self-adjoint matrices all the characteristic values of which are positive. Alternatively, they can be defined as matrices, all the hermitian quadratic forms of which are real and positive. It is well known that these two definitions are equivalent and it is the existence of two equivalent definitions which renders the subject particularly interesting. Thus, it follows from the first definition that A-1 is positive definite for positive definite A ; it follows from the second definition that the sum of two positive definite matrices is also positive definite. Similarly, we shall call a matrix positive semidefinite or, briefly, semidefinite if it is self-adjoint and none of its characteristic values are negative, or if all of its hermitian quadratic forms are non-negative. A matrix which is definite is also semidefinite but a semidefinite matrix is, of course, not necessarily definite.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 16 , 1964 , pp. 397 - 406
Copyright
Copyright © Canadian Mathematical Society 1964

References

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5. One of the writers (E.P.W.) wishes to use this opportunity to point out that Theorem I of his article On weakly positive matrices (Can. J. Math., 15 (1963), 313) is partially contained in O. Taussky's problem 4846 published in Am. Math. Monthly, 66 (1959), 429.Google Scholar
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