Remote Access Transactions of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] C. R. DePrima and C. R. Johnson, The range of 𝐴⁻¹𝐴* in 𝐺𝐿(𝑛,𝐶), Linear Algebra and Appl. 9 (1974), 209–222. MR 0361862
  • [2] Ky Fan, Generalized Cayley transforms and strictly dissipative matrices, Linear Algebra and Appl. 5 (1972), 155–172. MR 0296084
  • [3] Ky Fan, On real matrices with positive definite symmetric component, Linear and Multilinear Algebra 1 (1973), no. 1, 1–4. MR 0347857
  • [4] Charles R. Johnson, An inequality for matrices whose symmetric part is positive definite, Linear Algebra and Appl. 6 (1973), 13–18. MR 0311689
  • [5] A. M. Ostrowski and Olga Taussky, On the variation of the determinant of a positive definite matrix, Nederl. Akad. Wetensch. Proc. Ser. A. 54 = Indagationes Math. 13 (1951), 383–385. MR 0047597

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 15A45

Retrieve articles in all journals with MSC: 15A45

Additional Information

Keywords: Eigenvalues, Hadamard inequality, real part, positive definite
Article copyright: © Copyright 1975 American Mathematical Society