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Journal Article•DOI•

Vanishing Pseudo Schur Complements, Reverse Order Laws, Absorption Laws and Inheritance Properties

Kavita Bisht1, K. C. Sivakumar1•
Indian Institute of Technology Madras1
02 Jan 2018-Linear & Multilinear Algebra (Informa UK Limited)-Vol. 66, Iss: 1, pp 167-183
TL;DR: The problem of vanishing of a generalized Schur complement of a block matrix, implying that the other (generalized) Schur complements is zero, is revisited in this article, where absorption laws for two important classes of generalized inverses are considered.
Abstract: The problem of vanishing of a (generalized) Schur complement of a block matrix (corresponding to the leading principal subblock) implying that the other (generalized) Schur complement (corresponding to the trailing principal subblock) is zero, is revisited. Absorption laws for two important classes of generalized inverses are considered next. Inheritance properties of the generalized Schur complements in relation to the absorption laws are derived. Inheritance by the generalized principal pivot transform is also studied.
Citations
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Journal Article•DOI•
Absorption laws and reverse order laws for generalized core inverses

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Yuefeng Gao1, Jianlong Chen2, Long Wang3, Honglin Zou•
University of Shanghai for Science and Technology1, Southeast University2, Yangzhou University3
15 Mar 2021-Communications in Algebra
TL;DR: In this paper, the authors give necessary and sufficient conditions for the absorption laws and reverse order laws of two kinds of generalized core inverses to hold in general core-inverses.
Abstract: In this note, we give necessary and sufficient conditions for which the absorption laws and the reverse order laws of two kinds of generalized core inverses hold.

3 citations

References
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Journal Article•DOI•
Conditions for Positive and Nonnegative Definiteness in Terms of Pseudoinverses

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Arthur Albert
01 Mar 1969-Siam Journal on Applied Mathematics
TL;DR: Matrix pseudoinverses producing necessary and sufficient conditions for positive and nonnegative definiteness as discussed by the authors, and matrix pseudo-inverse producing necessary conditions for nonnegative determiniteness.
Abstract: Matrix pseudoinverses producing necessary and sufficient conditions for positive and nonnegative definiteness

398 citations

"Vanishing Pseudo Schur Complements,..." refers methods in this paper

  • ...The notion of Schur complement was extended to the case, where A−1 was replaced by the Moore–Penrose inverse of A, by Albert [1] who studied the positive definiteness and nonnegative definiteness for symmetric matrices using this formula....

    [...]

Journal Article•DOI•
Note on the generalized inverse of a matrix product.

[...]

T. N. E. Greville
01 Jan 1966-Siam Review

271 citations

Journal Article•DOI•
The reverse order law revisited

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Robert E. Hartwig1•
North Carolina State University1
01 Apr 1986-Linear Algebra and its Applications
TL;DR: In this article, necessary and sufficient conditions are given for the triple reverse order law (ABC) = C†B†A† to hold, and special cases are considered for some special cases.

88 citations

"Vanishing Pseudo Schur Complements,..." refers background in this paper

  • ...Hartwig studied this law for the product of three matrices and derived many characterizations for the formula (ABC)† = C†B†A† to hold [13]....

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  • ...This was simplified by Argirhiade who showed that the reverse law order holds if and only if R(A∗ABB∗) = R(BB∗A∗A) (see the references in [13])....

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Journal Article•DOI•
The absorption laws for the generalized inverses

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Xiaoji Liu1, Hongwei Jin1, Dragana S. Cvetković-Ilić2•
Guangxi University for Nationalities1, University of Niš2
01 Nov 2012-Applied Mathematics and Computation
TL;DR: The various types of mixed absorption law for the generalized inverses are considered and necessary and sufficient conditions for the absorption laws in terms of { 1 }, { 1 , 2 } , { 1, 3 } and { 1 -inverses} are given.

11 citations

"Vanishing Pseudo Schur Complements,..." refers background in this paper

  • ...We also refer the reader to the work reported in [7] where equivalent conditions for the absorption laws to hold, for the various generalized inverses of operators on Hilbert spaces....

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Journal Article•DOI•
A cancellation property of the moore–penrose inverse of triple products

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Tobias Damm1, Harald K. Wimmer2•
Kaiserslautern University of Technology1, University of Würzburg2
01 Feb 2009-Journal of The Australian Mathematical Society
TL;DR: In this paper, the authors studied the consistency of the matrix equation C(BXC) = X, where X denotes the Moore-Penrose inverse and BXC denotes the compliance matrix.
Abstract: We study the matrix equation C(BXC)†B = X†, where X† denotes the Moore-Penrose inverse. We derive conditions for the consistency of the equation and express all its solutions using singular vectors of B and C. Applications to compliance matrices in molecular dynamics, to mixed reverse-order laws of generalized inverses and to weighted Moore-Penrose inverses are given.

10 citations

"Vanishing Pseudo Schur Complements,..." refers background in this paper

  • ...and has been explored in [4], where the question of characterizing such an implication was left as an open problem....

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  • ...In [4], a question was asked as to whenM/D = 0 impliesM/A = 0....

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