Generalized reflexive and anti-reflexive solutions of $$AX=B$$ A X = B

Liu, Xifu; Yuan, Y.

doi:10.1007/s10092-014-0136-6

Cited by 9 publications

(6 citation statements)

References 9 publications

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“…In [10], the authors investigated the common (P, Q) generalized reflexive solution to (1.1). For the generalized reflexive and anti-reflexive solutions of AX = B, they were discussed in [3,4,11]. And for the generalized reflexive and anti-reflexive solutions of AX B = C, they were considered in [5,6].…”

Section: Introductionmentioning

confidence: 99%

The common (P, Q) generalized reflexive and anti-reflexive solutions to $$AX=B$$ A X = B and $$XC=D$$ X C = D

Liu

2015

Calcolo

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132

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In this paper, we establish some conditions for the existence and the representations for the common (P, Q) generalized reflexive and anti-reflexive solutions of matrix equations AX = B and XC = D, where P and Q are two generalized reflection matrices. Moreover, in corresponding solution set of the equations, the explicit expression of the nearest matrix to a given matrix in the Frobenius norm has been presented.

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Section: Introductionmentioning

confidence: 99%

The common (P, Q) generalized reflexive and anti-reflexive solutions to $$AX=B$$ A X = B and $$XC=D$$ X C = D

Liu

2015

Calcolo

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132

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“…Lemma 4.1. ( [21])Let A ∈ C k×m and B ∈ C k×n be given. Then (i) AX = B has a solution X ∈ C m×n r (P, Q) if and only if A 1 (P)A † 1 (P)B 1 (Q) = B 1 (Q).…”

Section: Properties Of the Reflexive And Anti-reflexive Solutions To Ax = Bmentioning

confidence: 99%

The extremal ranks and inertias of matrix expressions with respect to generalized reflexive and anti-reflexive matrices with applications

Liu¹

2018

Filomat

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The extremal ranks of matrix expressions of A-BXC and D-EYE*, and the extremal inertias of D-EYE* are discussed, where X and Y are reflexive (or anti-reflexive) and Hermitian reflexive (or anti-reflexive) matrices respectively. For the applications, we derive the extremal ranks of the reflexive and anti-reflexive solutions to AX = B. In addition, we also establish some conditions for the existence of common reflexive and anti-reflexive solutions to AX = B and CXD = E, and conditions for the solvability of some matrix equations and matrix inequalities.

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“…For the matrix equation AX � B, the Hermitian reflexive, antireflexive, generalized reflexive, and antireflexive solutions have been discussed in [15,16]. And different solution sets for other equations have also been discussed in [17][18][19][20].…”

Section: Introductionmentioning

confidence: 99%

Inverse Eigenvalue Problem and Least‐Squares Problem for Skew‐Hermitian {P,K + 1}‐Reflexive Matrices

Dong

2022

Journal of Mathematics

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This paper involves related inverse eigenvalue problem and least-squares problem of skew-Hermitian {P,k + 1}-reflexive(antireflexive) matrices and their optimal approximation problems. The above problems are studied by converting them into two simpler cases: k = 1 and k = 2. Firstly, with some special properties of skew-Hermitian {P,k + 1}-reflexive(antireflexive) matrices, the necessary and sufficient conditions for the solvability and the general solution are presented, and the solution of corresponding optimal approximation problems also given, respectively. Then, we give the least-squares solution of AX = B satisfying the special condition by the singular value decomposition. Finally, we give an algorithm and an example to illustrate our results.

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Generalized reflexive and anti-reflexive solutions of $$AX=B$$ A X = B

Cited by 9 publications

References 9 publications

The common (P, Q) generalized reflexive and anti-reflexive solutions to $$AX=B$$ A X = B and $$XC=D$$ X C = D

The common (P, Q) generalized reflexive and anti-reflexive solutions to $$AX=B$$ A X = B and $$XC=D$$ X C = D

The extremal ranks and inertias of matrix expressions with respect to generalized reflexive and anti-reflexive matrices with applications

Inverse Eigenvalue Problem and Least‐Squares Problem for Skew‐Hermitian {P,K + 1}‐Reflexive Matrices

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