The solution to matrix equation AX+XTC=B

Piao, Fengxian; Zhang, Qingling; Wang, Zhefeng

doi:10.1016/j.jfranklin.2007.05.002

Cited by 64 publications

(38 citation statements)

References 12 publications

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“…Then, by (15), x ij = −y j+1,i and, by (14), −y j+1,i = x i+1,j+1 , so we conclude that x ij = x i+1,j+1 . Hence, X is a Toeplitz matrix.…”

Section: Dimension Of the Solution Space For Pairs Of Blocksmentioning

confidence: 82%

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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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“…Then, by (15), x ij = −y j+1,i and, by (14), −y j+1,i = x i+1,j+1 , so we conclude that x ij = x i+1,j+1 . Hence, X is a Toeplitz matrix.…”

Section: Dimension Of the Solution Space For Pairs Of Blocksmentioning

confidence: 82%

“…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%

The solution of the equationAX + BX^⋆ = 0

Terán

2013

Linear and Multilinear Algebra

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“…In [40], the necessary and sufficient condition for the existence of the solution to the matrix equation (1.3) and its solution expression was investigated by the generalized inverse matrix. In [37], Cramer's rules for some quaternion matrix equations were obtained within the framework of the theory of the column and row determinants.…”

Section: Introductionmentioning

confidence: 99%

Two iterative algorithms for solving coupled matrix equations over reflexive and anti-reflexive matrices

Dehghan

Hajarian

2012

Comput. Appl. Math.

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Abstract. An n × n real matrix P is said to be a generalized reflection matrix if P T = P and P 2 = I (where P T is the transpose of P). A matrix A ∈ R n×n is said to be a reflexive (antireflexive) matrix with respect to the generalized reflection matrix P if A = P A P (A = −P A P). The reflexive and anti-reflexive matrices have wide applications in many fields. In this article, two iterative algorithms are proposed to solve the coupled matrix equationsover reflexive and anti-reflexive matrices, respectively. We prove that the first (second) algorithm converges to the reflexive (anti-reflexive) solution of the coupled matrix equations for any initial reflexive (anti-reflexive) matrix. Finally two numerical examples are used to illustrate the efficiency of the proposed algorithms.Mathematical subject classification: 15A06, 15A24, 65F15, 65F20.

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“…[13][14][15][16][17][18]26,[35][36][37][38][39][40][41][42][43][44][45][46]. The frequently studied Lyapunov equation BX + (BX ) * = A constitutes a major function in mathematics, such as optimal control, stability analysis, system theory, model reduction; see, e.g, [19,20]. The classical matrix equation CXC * + DY D * = A has been analyzed by numerous authors with different approaches; see, e.g.…”

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confidence: 99%