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A relation between the permanental and determinantal adjoints

Published online by Cambridge University Press:  09 April 2009

Marvin Marcus  and
Russell Merris
Marvin Marcus
Affiliation:
The University of CaliforniaSanta Barbara
Russell Merris
Affiliation:
The National Bureau of Standards Washington, U.S.A.
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Let Hn denote the set of complex n-square positive semidefinite hermtian matrices. We partially order Hn: If A, B, A–B єHn, write A > B. For A єHn, Write P(A) for the permanental adjoint of A, i.e., P(A) is the n-suare matrix whose i, j entry is per A(j/i), where A(j/i) is the submatrix of A obtained by deleting row j and column i. Now, P(A) is a principal submatrix of t he (n–1)st induced power matrix of AT. Hence, P(A) ∈Hn. Also D(A), the classical adjoing, is in Hn.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 15 , Issue 3 , May 1973 , pp. 270 - 271
Copyright
Copyright © Australian Mathematical Society 1973

References

1Djoković, D.Ž,‘On a conjecture by van der Waerden’,Mat. Vesnik (4)19(1967),272–276.Google Scholar
2Marcus, Marvin and Minc, Henryk‘Extensions of classical matrix inequalities’,Linear Algebra Appl.1(1968),421–444.CrossRefGoogle Scholar