四元数Lyapunov方程AX+XA~*=B的双自共轭解
|
摘要
【目的】研究四元数体上连续型Lyapunov方程AX+XA*=B的双自共轭解。【方法】利用双自共轭矩阵的结构特性及矩阵变换,将原问题转化为具有自共轭结构的方程问题,再通过自共轭矩阵的向量化刻画。【结果】获得了该方程存在双自共轭解的充要条件及通解表达式。【结论】所得结果扩展了Lyapunov方程的解形式,同时数值算例检验了所给算法的可行性。
[Purposes]To discuss dual self-conjugate solution of the quaternion Lyapunov equation AX+XA*=B.[Methods]The original problem is transformed into an equation problem with self-conjugate structure by using structural properties of dual self-conjugate matrix and matrix transformations.[Findings]An necessary and sufficient conditions for the existence of a dual self-conjugate solution and the general solution of the equation are obtained by the vectorization of the self-conjugate matrix.[Conclusions]The results expand solution forms of Lyapunov equation,and the numerical example demonstrate effectiveness of the proposed algorithm.
[Purposes]To discuss dual self-conjugate solution of the quaternion Lyapunov equation AX+XA*=B.[Methods]The original problem is transformed into an equation problem with self-conjugate structure by using structural properties of dual self-conjugate matrix and matrix transformations.[Findings]An necessary and sufficient conditions for the existence of a dual self-conjugate solution and the general solution of the equation are obtained by the vectorization of the self-conjugate matrix.[Conclusions]The results expand solution forms of Lyapunov equation,and the numerical example demonstrate effectiveness of the proposed algorithm.
引文
[1]SANCHES J M,MARQUES J S.Image denoising using the Lyapunov equation from non-uniform samples[C]//Image Analysis and Recognition:Third International Conference,Part I.Povoa de Varzim,Portugal:ICIAR,2006:351-358.
[2]PANASENKO E V,POKUTNYI O O.Boundary-value problems for the Lyapunov equation in Banach spaces[J].Journal of Mathematical Sciences,2017,223(3):298-304.
[3]BOGACHEV V I,ROCKNER M,SHAPOSHNIKOV SV.On existence of Lyapunov functions for a stationary Kolmogorov equation with a probability solution[J].Doklady Mathematics,2014,90(1):424-428.
[4]赵军,高岩.二阶线性系统族的共同二次Lyapunov函数[J].重庆师范大学学报(自然科学版),2012,29(4):52-56.ZHAO J,GAO Y.Common quadratic Lyapunov functions for second order linear systems[J].Journal of Chongqing Normal University(Natural Science),2012,29(4):52-56.
[5]黄敬频.一类混合型Lyapunov方程的对称正定解[J].工程数学学报,2008,25(2):313-320.HUANG J P.The symmetric positive definite solutions of a class of mixed-type Lyapunov matrix equations[J].Journal of Engineering Mathematics,2008,25(2):313-320.
[6]邓勇,黄敬频.四元数体上离散型Lyapunov方程的反问题解[J].西南师范大学学报(自然科学版),2015,40(7):1-6.DENG Y,HUANG J P.On inverse problem of discrete Lyapunov equation over quaternion field[J].Journal of Southwest Normal University(Natural Science),2015,40(7):1-6.
[7]SUN H,ZHANG J.Solving Lyapunov equation by quantum algorithm[J].Control Theory Technology,2017,15(4):267-273.
[8]胡锡炎,张磊,谢冬秀.双对称矩阵逆特征值问题解存在的条件[J].计算数学,1998,20(4):409-418.HU X Y,ZHANG L,XIE D X.The solvability conditions for the inverse eigenvalue problems of bisymmetric matrices[J].Computational Mathematics,1998,20(4):409-418.
[9]YUANG S F,WANG Q W,YU Y B.On Hermitian solutions of the split quaternion matrix equation AXB+CXD=E[J].Advances in Applied Clifford Algebras,2017,27(4):3235-3252.
[2]PANASENKO E V,POKUTNYI O O.Boundary-value problems for the Lyapunov equation in Banach spaces[J].Journal of Mathematical Sciences,2017,223(3):298-304.
[3]BOGACHEV V I,ROCKNER M,SHAPOSHNIKOV SV.On existence of Lyapunov functions for a stationary Kolmogorov equation with a probability solution[J].Doklady Mathematics,2014,90(1):424-428.
[4]赵军,高岩.二阶线性系统族的共同二次Lyapunov函数[J].重庆师范大学学报(自然科学版),2012,29(4):52-56.ZHAO J,GAO Y.Common quadratic Lyapunov functions for second order linear systems[J].Journal of Chongqing Normal University(Natural Science),2012,29(4):52-56.
[5]黄敬频.一类混合型Lyapunov方程的对称正定解[J].工程数学学报,2008,25(2):313-320.HUANG J P.The symmetric positive definite solutions of a class of mixed-type Lyapunov matrix equations[J].Journal of Engineering Mathematics,2008,25(2):313-320.
[6]邓勇,黄敬频.四元数体上离散型Lyapunov方程的反问题解[J].西南师范大学学报(自然科学版),2015,40(7):1-6.DENG Y,HUANG J P.On inverse problem of discrete Lyapunov equation over quaternion field[J].Journal of Southwest Normal University(Natural Science),2015,40(7):1-6.
[7]SUN H,ZHANG J.Solving Lyapunov equation by quantum algorithm[J].Control Theory Technology,2017,15(4):267-273.
[8]胡锡炎,张磊,谢冬秀.双对称矩阵逆特征值问题解存在的条件[J].计算数学,1998,20(4):409-418.HU X Y,ZHANG L,XIE D X.The solvability conditions for the inverse eigenvalue problems of bisymmetric matrices[J].Computational Mathematics,1998,20(4):409-418.
[9]YUANG S F,WANG Q W,YU Y B.On Hermitian solutions of the split quaternion matrix equation AXB+CXD=E[J].Advances in Applied Clifford Algebras,2017,27(4):3235-3252.