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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

Inequalities for a complex matrix whose real part is positive definite

Author: Charles R. Johnson
Journal: Trans. Amer. Math. Soc. 212 (1975), 149-154
MSC: Primary 15A45
MathSciNet review: 0424851
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Abstract: Denote the real part of $ A \in {M_n}(C)$ by $ H(A) = {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}(A + {A^\ast})$. We provide dual inequalities relating $ H({A^{ - 1}})$ and $ H{(A)^{ - 1}}$ and an identity between two functions of A when A satisfies $ H(A) > 0$. As an application we give an inequality (for matrices A satisfying $ H(A) > 0$) which generalizes Hadamard's determinantal inequality for positive definite matrices.

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PII: S 0002-9947(1975)0424851-2
Keywords: Eigenvalues, Hadamard inequality, real part, positive definite
Article copyright: © Copyright 1975 American Mathematical Society