A Note on the<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M1"><mml:mrow><mml:mi>⊤</mml:mi></mml:mrow></mml:math>-Stein Matrix Equation

Chiang, Chun Yueh

doi:10.1155/2013/824641

Cited by 8 publications

(3 citation statements)

References 16 publications

(37 reference statements)

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“…(3)? The recent examples in the literature [1,5] and Example 5.2 tell us that they are almost the same. What properties of the matrix operator f such that the conditions of uniquely solvable of Eq.…”

Section: Concluding Remarkmentioning

confidence: 78%

Thirty-Year Outcomes of the National Hepatitis B Immunization Program in Taiwan

Chiang

Yang²,

You

et al. 2013

JAMA

207

118

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In this paper, we derive a formula to compute the solution of the linear matrix equation X = Af (X)B + C via finding any solution of a specific Stein matrix equation X = AX B + C, where the linear (or anti-linear) matrix operator f is period-n. According to this formula, we should pay much attention to solve the Stein matrix equation from recently famous numerical methods. For instance, Smith-type iterations, Bartels-Stewart algorithm, and etc.. Moreover, this transformation is used to provide necessary and sufficient conditions of the solvable of the linear matrix equation. On the other hand, it can be proven that the general solution of the linear matrix equation can be presented by the general solution of the Stein matrix equation. The necessary condition of the uniquely solvable of the linear matrix equation is developed. It is shown that several representations of this formula are coincident. Some examples are presented to illustrate and explain our results.

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Section: Concluding Remarkmentioning

confidence: 78%

Thirty-Year Outcomes of the National Hepatitis B Immunization Program in Taiwan

Chiang

Yang²,

You

et al. 2013

JAMA

207

118

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“…Matrix equations of this form are sometimes called T-Stein equations [10]. The following theorem shows that the iterates produced by the fixed point iteration applied to (6.1) are contained in the tensor product V m ⊗ W m of the block Krylov subspaces in (4.3) and (4.4).…”

Section: Relation To a T-stein Equation And A Fixed Point Iterationmentioning

confidence: 99%

Projection methods for large-scale T-Sylvester equations

et al. 2016

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The matrix Sylvester equation for congruence, or T-Sylvester equation, has recently attracted considerable attention as a consequence of its close relation to palindromic eigenvalue problems. The theory concerning T-Sylvester equations is rather well understood and there are stable and efficient numerical algorithms which solve these equations for small-to medium-sized matrices. However, developing numerical algorithms for solving large-scale T-Sylvester equations still remains an open problem. In this paper, we present several projection algorithms based on different Krylov spaces for solving this problem when the right-hand side of the T-Sylvester equation is a low-rank matrix. The new algorithms have been extensively tested, and the reported numerical results show that they work very well in practice, offering a clear guidance on which algorithm is the most convenient in each situation.

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“…It was discovered by Hamilton in 1843, and has many applications in recent research areas such as computer graphics, control theory, and signal processing (see, e.g., [2,6,7,8,17,18,21,22]). One of the research directions in this area is solving matrix equations over H. In the literature, the generalized Sylvester matrix equation AX − XB = C and Stein matrix equation X − AX B = C have been well studied over real numbers R and complex numbers C due to many applications in control theory (see, e.g., [1,3,4,5,13,16,20,25,27,30]). Because of the non-commutative algebraic structure of H, solving matrix equations over H appears more challenging and has attracted more and more attentions recently.…”

mentioning

confidence: 99%

Consistency of Quaternion Matrix Equations $AX^{\star}-XB=C$ and $X-AX^\star B=C$

Liu

Wang

Zhang

2019

ELA

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For a given ordered units triple $\{q_1, q_2, q_3\}$, the solutions to the quaternion matrix equations $AX^{\star}-XB=C$ and $X-AX^{\star}B=C$, $X^{\star} \in \{ X , X^{\eta} , X^* , X^{\eta*}\}$, where $X^*$ is the conjugate transpose of $X$, $X^{\eta}=-\eta X \eta$ and $X^{\eta*}=-\eta X^* \eta$, $\eta \in \{q_1, q_2, q_3\}$, are discussed. Some new real representations of quaternion matrices are used, which enable one to convert $\eta$-conjugate (transpose) matrix equations into some real matrix equations. By using this idea, conditions for the existence and uniqueness of solutions to the above quaternion matrix equations are derived. Also, methods to construct the solutions from some related real matrix equations are presented.

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A Note on the⊤-Stein Matrix Equation

Cited by 8 publications

References 16 publications

Thirty-Year Outcomes of the National Hepatitis B Immunization Program in Taiwan

Thirty-Year Outcomes of the National Hepatitis B Immunization Program in Taiwan

Projection methods for large-scale T-Sylvester equations

Consistency of Quaternion Matrix Equations $AX^{\star}-XB=C$ and $X-AX^\star B=C$

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