The matrix equation AX − YB = C

Baksalary, Jerzy K.; Kala, Radosław

doi:10.1016/0024-3795(79)90004-1

Cited by 78 publications

(23 citation statements)

References 2 publications

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“…The first question arising in examinating the above linear equation is about its solvability. An answer to this question can be found in [16] (see also [3], [14]); we report it in the following.…”

Section: Then the Newton Iteration For (8) Is As Followsmentioning

confidence: 87%

A Hybrid Numerical Technique for the Solution of a Class of Implicit Matrix Differential Equation

Buono

Lopez

2004

Computational Science - ICCS 2004

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Abstract. This paper is concerned with the numerical solution of an implicit matrix differential system of the form Y TẎ − F (t, Y ) = 0, where Y (t) is a n × n real matrix which may converge to a singular matrix. We propose a hybrid numerical technique based on an implicit second order Runge Kutta scheme which derives a particular algebraic Riccati equation and via its solution approximates the solutions of the differential problem at hand. Numerical examples demonstrating the behavior of the proposed approach are also reported.

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Section: Then the Newton Iteration For (8) Is As Followsmentioning

confidence: 87%

A Hybrid Numerical Technique for the Solution of a Class of Implicit Matrix Differential Equation

Buono

Lopez

2004

Computational Science - ICCS 2004

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“…The factorizations used in Theorem 5 allow us to give the explicit general solution of BCX + YAB = B for A ∈ C m×n , B ∈ C n×n , and C ∈ C n×q . Indeed, by using the expressions of A, B, and C given in Theorem 5 it is not hard to show that BCX+YAB = B is consistent (in the unknowns X and Y) if and only if (I − C 1 C − 1 )(I − A − 1 A 1 ) = O (see [1,3]). Here, A − denotes a {1}-generalized inverse of A (that is, AA − A = A).…”

Section: Remarkmentioning

confidence: 99%

“…for arbitrary matrices Z, W, M, and N. Notice that matrices of smaller sizes are used in our computations compared to that given in [1].…”

Section: Remarkmentioning

confidence: 99%

Inequalities and equalities for ℓ = 2 (Sylvester), ℓ = 3 (Frobenius), and ℓ > 3 matrices

Thome

2016

Aequat. Math.

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Thome, N. (2016). Inequalities and equalities for l = 2 (Sylvester), l = 3 (Frobenius), and l > 3 matrices. Aequationes Mathematicae. 90 (5) AbstractThis paper gives simple proofs of Sylvester ( = 2) and Frobenius ( = 3) inequalities. Moreover, a new sufficient condition for the equality of the Frobenius inequality is provided. In addition, an extension for > 3 matrices and an application to generalized inverses are provided.

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“…Since R(A) ∩ R(B) ⊆ R(C), we have R((AXA * ) 1 2 ) ⊆ R(C) and R((BY B * ) 1 2 ) ⊆ R(C). Noting to R(AXA * ) ⊆ R((AXA * ) 1 2 ) and R(BY B * ) ⊆ R((BY B * ) 1 2 ), we obtain R(AXA * + BY B * ) ⊆ R(AXA * ) + R(BY B * ) ⊆ R(C). (3.14)…”

Section: In This Case There Exists a Unique Operator X Satisfying R(mentioning

confidence: 99%

Operator equations AX+YB=C and AXA⁎+BYB⁎=C in Hilbert C⁎-modules

Mousavi

Eskandari

Moslehian

et al. 2017

Linear Algebra and its Applications

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Abstract. Let A, B and C be adjointable operators on a Hilbert C * -module E . Giving a suitable version of the celebrated Douglas theorem in the context of Hilbert C * -modules, we present the general solution of the equation AX + Y B = C when the ranges of A, B and C are not necessarily closed. We examine a result of Fillmore and Williams in the setting of Hilbert C * -modules. Moreover, we obtain some necessary and sufficient conditions for existence of a solution for AXA * + BY B * = C. Finally, we deduce that there exist nonzero operators X, Y ≥ 0 and Z such that AXA * + BY B * = CZ, when A, B and C are given subject to some conditions. Introduction and PreliminariesRecently several operator equations have been extended from matrices to infinite dimensional spaces, i.e., Hilbert spaces and Hilbert C * -modules; see [11] and references therein. Recall that the notion of Hilbert C * -module is a natural generalization of that of Hilbert space arising by replacing the field of scalars C by a C * -algebra. Generalized inverses are useful tools for investigation of solutions of operator equations in the setting of Hilbert C * -modules but these inverses need the strong condition of closedness of ranges of considered operators. Fang et al. [6,7] have studied the solvability of operator equations without the closedness condition on ranges of operators by employing a generalization of a known theorem of Douglas [4, Theorem 1] in the framework of Hilbert C * -modules. In their results, concentration is based on the idea of using more general (orthogonal) projections instead of projections such as AA † . They investigated the equations AX = B Inspired by Fang et al., we investigate the solution of equations AX + Y B = C and AXA * +BY B * = C without the condition of closedness of ranges. This paper is organized as follows. First, we recall some basic information about Hilbert C * -modules. In Section 2, we present an example that shows that the conditions CC * ≤ λAA * for some λ > 0 and R(C) ⊆ R(A) are not equivalent in the setting 2010 Mathematics Subject Classification. 15A24, 46L08, 47A05, 47A62.

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The matrix equation AX − YB = C

Cited by 78 publications

References 2 publications

A Hybrid Numerical Technique for the Solution of a Class of Implicit Matrix Differential Equation

A Hybrid Numerical Technique for the Solution of a Class of Implicit Matrix Differential Equation

Inequalities and equalities for ℓ = 2 (Sylvester), ℓ = 3 (Frobenius), and ℓ > 3 matrices

Operator equations AX+YB=C and AXA⁎+BYB⁎=C in Hilbert C⁎-modules

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