Inequalities for a complex matrix whose real part is positive definite

Johnson, Charles R.

doi:10.1090/S0002-9947-1975-0424851-2

Inequalities for a complex matrix whose real part is positive definite
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by Charles R. Johnson

Trans. Amer. Math. Soc. 212 (1975), 149-154

DOI: https://doi.org/10.1090/S0002-9947-1975-0424851-2

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Abstract:

Denote the real part of $A \in {M_n}(C)$ by $H(A) = 1/2(A + {A^\ast })$. 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

C. R. DePrima and C. R. Johnson, The range of $A^{-1}A^{\ast }$ in $\textrm {GL}(n,\,C)$, Linear Algebra Appl. 9 (1974), 209–222. MR 361862, DOI 10.1016/0024-3795(74)90039-1
Ky Fan, Generalized Cayley transforms and strictly dissipative matrices, Linear Algebra Appl. 5 (1972), 155–172. MR 296084, DOI 10.1016/0024-3795(72)90025-0
Ky Fan, On real matrices with positive definite symmetric component, Linear and Multilinear Algebra 1 (1973), no. 1, 1–4. MR 347857, DOI 10.1080/03081087308817001
Charles R. Johnson, An inequality for matrices whose symmetric part is positive definite, Linear Algebra Appl. 6 (1973), 13–18. MR 311689, DOI 10.1016/0024-3795(73)90003-7
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