A shifted complex global Lanczos method and the quasi-minimal residual variant for the Stein-conjugate matrix equation <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll" id="d1e418" altimg="si3.gif"><mml:mi>X</mml:mi><mml:mo>+</mml:mo><mml:mi>A</mml:mi><mml:mover accent="false" class="mml-overline"><mml:mrow><mml:mi>X</mml:mi></mml:mrow><mml:mo accent="true">¯</mml:mo></mml:mover><mml:mi>B</mml:mi><mml:mo>=</mml:mo><mml:mi>C</mml:mi></mml:math>

Li, Shengkun; Huang, Ting‐Zhu

doi:10.1016/j.cam.2019.02.009

Cited by 8 publications

(2 citation statements)

References 43 publications

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“…The matrix equation AX − XB = C, where X denotes the matrix obtained by taking the complex conjugate of each element of X, is called the normal Sylvester-conjugate matrix equation. The matrix equation X − AXF = C is called the Kalman-Yakubovich-conjugate matrix equation (also known as the Stein-conjugate matrix equation, see [10]). The matrix equation X − AXF = BY is called the Yakubovich-conjugate matrix equation ( [11,12]).…”

Section: Introductionmentioning

confidence: 99%

A General Approach to Sylvester-Polynomial-Conjugate Matrix Equations

Mazurek

2024

Symmetry

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Sylvester-polynomial-conjugate matrix equations unify many well-known versions and generalizations of the Sylvester matrix equation AX−XB=C which have a wide range of applications. In this paper, we present a general approach to Sylvester-polynomial-conjugate matrix equations via groupoids, vector spaces, and matrices over skew polynomial rings. The obtained results are applied to Sylvester-polynomial-conjugate matrix equations over complex numbers and quaternions. The main role in our approach is played by skew polynomial rings, which are well-known tools in algebra to provide examples of asymmetry between left-sided and right-sided versions of many ring objects.

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Section: Introductionmentioning

confidence: 99%

A General Approach to Sylvester-Polynomial-Conjugate Matrix Equations

Mazurek

2024

Symmetry

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“…Recent work on Sylvester equation and its generalizations can be found in [20]- [28] and the references therein. A generalized version of the Stein matrix equations XF − AX = C, X − AXF = C where X denotes the conjugate of X, was investigated in [29], [30] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

An eigen decomposition based closed-form solution for the Discrete Lyapunov and Stein Equations

Patel¹,

Khan²

2021

Preprint

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A simple closed-form solution to the discrete Lyapunov equation (DLE) is established for certain families of matrices. This solution is expressed in terms of the eigen decomposition (ED) for which closed-form solutions are known for all 2 × 2, 3 × 3 and certain families of matrices. For general matrices, the proposed ED based closed-form solution can be used as an efficient numerical solution when the ED can be computed. The result is then extended to give closed-form solutions for a generalization of the DLE, called the Stein equation. The proposed explicit solution's complexity is of the same order as iterative solutions and significantly smaller than known closed-form solutions. These solutions may prove convenient for analysis and synthesis problems related to these equations due to their compact form.

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Nonlinear waves behaviors for a coupled generalized nonlinear Schrödinger–Boussinesq system in a homogeneous magnetized plasma

Lan

Guo

2020

Nonlinear Dyn

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A shifted complex global Lanczos method and the quasi-minimal residual variant for the Stein-conjugate matrix equation X+AX¯B=C

Cited by 8 publications

References 43 publications

A General Approach to Sylvester-Polynomial-Conjugate Matrix Equations

A General Approach to Sylvester-Polynomial-Conjugate Matrix Equations

An eigen decomposition based closed-form solution for the Discrete Lyapunov and Stein Equations

Nonlinear waves behaviors for a coupled generalized nonlinear Schrödinger–Boussinesq system in a homogeneous magnetized plasma

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