On löwner-ordering antitonicity of matrix inversion

Liski, Erkki P.

doi:10.1007/bf02029073

Cited by 3 publications

(2 citation statements)

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“…The partial inertia of difference of two Hermitian generalized inverses of Hermitian matrices of the same size, as well as the Löwner partial ordering of Hermitian generalized inverses of two Hermitian matrices were studied by some authors; see, e.g., [2,3,4,11,12,13,18,24,36,37]. Setting P = I m in Theorem 3.1, we obtain the following results on the partial inertia of A − − B − and their consequences.…”

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confidence: 85%

On an equality and four inequalities for generalized inverses of Hermitian matrices

Tian¹

2012

ELA

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Abstract. A Hermitian matrix X is called a g-inverse of a Hermitian matrix A, denoted by A − , if it satisfies AXA = A. In this paper, a group of explicit formulas are established for calculating the global maximum and minimum ranks and inertias of the difference A − − P N − P * , where both A − and N − are Hermitian g-inverses of two Hermitian matrices A and N , respectively. As a consequence, necessary and sufficient conditions are derived for the matrix equality A − = P N − P * to hold, and the four matrix inequalities A − > (≥, <, ≤) P N − P * in the Löwner partial ordering to hold, respectively. In addition, necessary and sufficient conditions are established for the Hermitian matrix equality A † = P N † P * to hold, and the four Hermitian matrix inequalities A † > (≥, < , ≤) P N † P * to hold, respectively, where (·) † denotes the Moore-Penrose inverse of a matrix. As applications, identifying conditions are given for the additive decomposition of a Hermitian g-inverse C − = A − + B − (parallel sum of two Hermitian matrices) to hold, as well as the four matrix inequalities C − > (≥, <, ≤) A − + B − in the Löwner partial ordering to hold, respectively.

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confidence: 85%

On an equality and four inequalities for generalized inverses of Hermitian matrices

Tian¹

2012

ELA

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“…To do this we need some auxiliary results. First, it is clear from Corollary 16 and the statement of Theorem [56,39], [63,Chap. 8]) is a satisfactory answer for our purposes.…”

Section: Corollary 16 If Amentioning

confidence: 95%

Matrix monotonicity and concavity of the principal pivot transform

Beard¹,

Welters²

2023

Preprint

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We prove the (generalized) principal pivot transform is matrix monotone, in the sense of the Löwner ordering, under minimal hypotheses. This improves on the recent results of J. E. Pascoe and R. Tully-Doyle, Monotonicity of the principal pivot transform, Linear Algebra Appl. 662 (2022) in two ways. First, we use the "generalized" principal pivot transform, where matrix inverses in the classical definition of the principal pivot transform are replaced with Moore-Penrose pseudoinverses. Second, the hypotheses on matrices for which monotonicity holds is relaxed and, in particular, we find the weakest hypotheses possible for which it can be true. We also prove the principal pivot transform is a matrix convex function on positive semi-definite matrices that have the same kernel (and, in particular, on positive definite matrices). Our proof is a corollary of a minimization variational principle for the principal pivot transform.

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Matrix monotonicity and concavity of the principal pivot transform

Beard,

Welters

2024

Linear Algebra and its Applications

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On löwner-ordering antitonicity of matrix inversion

Cited by 3 publications

References 9 publications

On an equality and four inequalities for generalized inverses of Hermitian matrices

On an equality and four inequalities for generalized inverses of Hermitian matrices

Matrix monotonicity and concavity of the principal pivot transform

Matrix monotonicity and concavity of the principal pivot transform

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