On the reflexive solutions of the matrix equation AXB+CYD=E
1
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
A matrix P∈C^(n×n) is called a generalized reflection matrix if P^(*) and P^(2)=I. An n×n complex matrix A is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A=PAP (A=-PAP). It is well-known that the reflexive and anti-reflexive matrices with respect to the generalized reflection matrix P have many special properties and widely used in engineering and scientific computations. In this paper, we give new necessary and sufficient conditions for the existence of the reflexive (anti-reflexive) solutions to the linear matrix equation AXB+CYD=E and derive representation of the general reflexive (anti-reflexive) solutions to this matrix equation. By using the obtained results, we investigate the reflexive (anti-reflexive) solutions of some special cases of this matrix equation.
Abstract
A matrix P∈C^(n×n) is called a generalized reflection matrix if P^(*) and P^(2)=I. An n×n complex matrix A is said to be a reflexive (anti-reflexive) matrix with respect to the generalized reflection matrix P if A=PAP (A=-PAP). It is well-known that the reflexive and anti-reflexive matrices with respect to the generalized reflection matrix P have many special properties and widely used in engineering and scientific computations. In this paper, we give new necessary and sufficient conditions for the existence of the reflexive (anti-reflexive) solutions to the linear matrix equation AXB+CYD=E and derive representation of the general reflexive (anti-reflexive) solutions to this matrix equation. By using the obtained results, we investigate the reflexive (anti-reflexive) solutions of some special cases of this matrix equation.
참고문헌 (31)
- J. K. Baksalary, "The matrix equation AX−Y B = C", Linear Algebra Appl, 25, pp.41~43 (1979)
- J. K. Baksalary, "The matrix equation AXB + CY D = E", Linear Algebra Appl, 30, pp.141~147 (1980)
- H. C. Chen, "Generalized reflexive matrices: special properties and applications", SIAM J. Matrix Anal. Appl., 19(1), pp.140~153 (1998)
- The SAS Domain Decomposition Method for Structural Analysis, CSRD Teach, "H. C. Chen", Center for Supercomputing Research and Development, University of Illinois, ~, pp.754 (1988)
- Numerical linear algebra algorithms on the ceder system in Parrallel computations and Their Impact on Mechanics, "H. C. Chen", The American Society of Mechanical Engineers, 101~125, pp.86 (1987)
- K. E. Chu, "Singular value and generalized singular value decompositions and the solution of linear matrix equations", Linear Algebra Appl, 88, pp.83~98 (1987)
- D. S. Cvetkovi´c-Ili´c, "The reflexive solutions of the matrix equation AXB = C", Comput. Math. Appl, 51(6), pp.897~902 (2006)
- Dragana S. Cvetkovic-Ilic, "The solutions of some operator equations", 대한수학회지, 45(5), pp.1417~1425 (2008), 대한수학회
- D. S. Cvetkovi´c-Ili´c, "Re-nnd solutions of the matrix equation AXB = C", J. Aust. Math. Soc, 84(1), pp.63~72 (2008)
- M. Dehghan, "An iterative algorithm for the reflexive solutions of the generalized coupled Sylvester matrix equations and its optimal approximation", Appl. Math. Comput, 202(2), pp.571~588 (2008)
- M. Dehghan, "An iterative algorithm for solving a pair of matrix equations AY B = E, CY D = F over generalized centro-symmetric matrices", Comput. Math. Appl, 56(12), pp.3246~3260 (2008)
- M. Dehghan, "The reflexive and anti-reflexive solutions of a linear matrix equation and systems of matrix equations", Rocky Mountain J. Math.
- M. Dehghan, "Finite iterative algorithms for the reflexive and anti-reflexive solutions of the matrix equation A1X1B1 + A2X2B2 = C", Math. Comput. Model, 49(9), pp.1937~1959 (2009)
- F. Ding, "Gradient based iterative algorithms for solving a class of matrix equations", IEEE Trans. Automat. Control, 50(8), pp.1216~1221 (2005)
- F. Ding, "Hierarchical gradient-based identification of multivariable discrete-time systems", Automatica J. IFAC, 41(2), pp.315~325 (2005)
- F. Ding, "Iterative least-squares solutions of coupled Sylvester matrix equations", Systems Control Lett, 54(2), pp.95~107 (2005)
- F. Ding, "Hierarchical least squares identification methods for multivariable systems", IEEE Trans. Automat. Control, 50(3), pp.397~402 (2005)
- F. Ding, "On iterative solutions of general coupled matrix equations", SIAM J. Control Optim, 44(6), pp.2269~2284 (2006)
- F. Ding, "Iterative solutions of the generalized Sylvester matrix equations by using the hierarchical identification principle", Appl. Math. Comput, 197(1), pp.41~50 (2008)
- H. Flanders, "On the matrix equations AX−XB = C and AX−Y B = C", SIAM J. Appl. Math, 32(4), pp.707~710 (1977)
- Z. Y. Peng, "The reflexive and anti-reflexive solutions of the matrix equation AX = B", Linear Algebra Appl, 375, pp.147~155 (2003)
- W. E. Roth, "The equations AX−Y B = C and AX−XB = C in matrices", Proc. Amer. Math. Soc, 3, pp.392~396 (1952)
- Q. W. Wang, "A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity", Linear Algebra Appl, 384, pp.43~54 (2004)
- Q. W. Wang, "A system of four matrix equations over von Neumann regular rings and its applications", Acta Math. Sin. (Engl. Ser.), 21(2), pp.323~334 (2005)
- Q. W. Wang, "Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations", Comput. Math. Appl, 49(5), pp.641~650 (2005)
- Q. W. Wang, "The common solution to six quaternion matrix equations with applications", Appl. Math. Comput, 198(1), pp.209~226 (2008)
- Q. W. Wang, "Ranks and the least-norm of the general solution to a system of quaternion matrix equations" (2008), Linear Algebra Appl
- Q. W.Wang, "The reflexive re-nonnegative definite solution to a quaternion matrix equation", Electron. J. Linear Algebra, 17, pp.88~101 (2008)
- G. Xu, "On solutions of matrix equation AXB + CY D = F", Linear Algebra Appl, 279, pp.93~109 (1998)
- B. Zhou, "Solutions to generalized Sylvester matrix equation by Schur decomposition", Internat. J. Systems Sci, 38(5), pp.369~375 (2007)
- B. Zhou, "On the generalized Sylvester mapping and matrix equations", Systems Control Lett, 57(3), pp.200~208 (2008)