The solution of the equation<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:mrow><mml:mi mathvariant="italic">AX</mml:mi><mml:mo>+</mml:mo><mml:msup><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mrow><mml:mi>★</mml:mi></mml:mrow></mml:msup><mml:mi>B</mml:mi><mml:mo>=</mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>

Terán, Fernando De; Dopico, Froilán M.; Guillery, Nathan; Montealegre, Daniel; Reyes, Nicolás

doi:10.1016/j.laa.2012.11.014

Cited by 28 publications

(29 citation statements)

References 17 publications

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“…Now, since A and B commute and at least one of A and B is nilpotent, we have that AB is also nilpotent. Then the result is an immediate consequence of Lemma 4 in [7].…”

Section: Proof Of the Dimension Count Expression For The Solutionmentioning

confidence: 65%

“…This way, both the results and techniques contained in the present paper look like very much to the ones in [7], where the homogeneous -Sylvester equation is solved. Let us mention one interesting similarity between both equations, which is in the basis of the procedure followed to solve them.…”

Section: Introductionmentioning

confidence: 73%

“…It is worth to compare theorems 3.1 and 3.2 with theorems 3 and 4 in [7]. In that case, the dimension of the solution space for both AX + X B = 0 depends on the KCF of A + λB , which may be different depending on whether = T or = * .…”

Section: Dimension Of the Solution Spacementioning

confidence: 99%

“…All these problems have been addressed (and, essentially, solved) by different authors for the general -Sylvester equation or the homogeneous case AX + X B = 0 of this equation [1,2,6,7,12,14,18].…”

Section: Introductionmentioning

confidence: 99%

“…Let us mention one interesting similarity between both equations, which is in the basis of the procedure followed to solve them. On the one hand, the solutions of AX + X B = 0 are in one-to-one linear correspondence with the solutions of AX +X B = 0, where A+λ B is a matrix pencil strictly equivalent to A + λB [7,Theorem 1]. Hence, the solution of AX + X B = 0 depends only on the Kronecker canonical form (KCF) of A + λB , and on two nonsingular matrices leading A + λB to its KCF.…”

Section: Introductionmentioning

confidence: 99%

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The solution of the equationAX + BX^⋆ = 0

Terán

2013

Linear and Multilinear Algebra

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We give a complete solution of the matrix equation AX + BX = 0, where A, B ∈ C m×n are two given matrices, X ∈ C n×n is an unknown matrix, and denotes the transpose or the conjugate transpose. We provide a closed formula for the dimension of the solution space of the equation in terms of the Kronecker canonical form of the matrix pencil A + λB, and we provide also an expression for the solution X in terms of this canonical form, together with two invertible matrices leading A + λB to the canonical form by strict equivalence.

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“…Now, since A and B commute and at least one of A and B is nilpotent, we have that AB is also nilpotent. Then the result is an immediate consequence of Lemma 4 in [7].…”

Section: Proof Of the Dimension Count Expression For The Solutionmentioning

confidence: 65%

Section: Introductionmentioning

confidence: 73%

Section: Dimension Of the Solution Spacementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The solution of the equationAX + BX^⋆ = 0

Terán

2013

Linear and Multilinear Algebra

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Nonsingular systems of generalized Sylvester equations: An algorithmic approach

Terán

Iannazzo

Poloni

et al. 2019

Numerical Linear Algebra App

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Summary We consider the uniqueness of solution (i.e., nonsingularity) of systems of r generalized Sylvester and ⋆‐Sylvester equations with n×n coefficients. After several reductions, we show that it is sufficient to analyze periodic systems having, at most, one generalized ⋆‐Sylvester equation. We provide characterizations for the nonsingularity in terms of spectral properties of either matrix pencils or formal matrix products, both constructed from the coefficients of the system. The proposed approach uses the periodic Schur decomposition and leads to a backward stable O(n3r) algorithm for computing the (unique) solution.

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On the Consistency of the Matrix Equation $$X^\top A X=B$$ when B is Symmetric

2021

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We provide necessary and sufficient conditions for the matrix equation X J AX " B to be consistent when B is a symmetric matrix, for all matrices A with a few exceptions. The matrices A, B, and X (unknown) are matrices with complex entries. We first see that we can restrict ourselves to the case where A and B are given in canonical form for congruence and, then, we address the equation with A and B in such form. The characterization strongly depends on the canonical form for congruence of A. The problem we solve is equivalent to: given a complex bilinear form (represented by A) find the maximum dimension of a subspace such that the restriction of the bilinear form to this subspace is a symmetric non-degenerate bilinear form.

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The solution of the equationAX+X★B=0

Cited by 28 publications

References 17 publications

The solution of the equationAX + BX^⋆ = 0

The solution of the equationAX + BX^⋆ = 0

Nonsingular systems of generalized Sylvester equations: An algorithmic approach

On the Consistency of the Matrix Equation $$X^\top A X=B$$ when B is Symmetric

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The solution of the equationAX+X★B=0

Cited by 28 publications

References 17 publications

The solution of the equationAX + BX⋆ = 0

The solution of the equationAX + BX⋆ = 0

Nonsingular systems of generalized Sylvester equations: An algorithmic approach

On the Consistency of the Matrix Equation $$X^\top A X=B$$ when B is Symmetric

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Product

Resources

About

The solution of the equationAX + BX^⋆ = 0

The solution of the equationAX + BX^⋆ = 0