The anti-symmetric ortho-symmetric solution of a linear matrix equation and its optimal approximation
Tóm tắt
This paper discusses the anti-symmetric ortho-symmetric solution of a linear matrix equation and its optimal approximation. By the generalized singular value decomposition of the matrices, the necessary and sufficient conditions for the solvability of the matrix equation and the general form of the anti-symmetric ortho-symmetric solution are given. In addition, the existence and uniqueness of the optimal approximation are proved. Numerical methods of the optimal approximation to a given matrix and numerical experiments are described.
Tài liệu tham khảo
Dai, H., Lancaster, P.: Linear matrix equation from an inverse problem of vibration theory. Linear Algebra Appl. 246, 31–47 (1996)
Peng, Y.X., Li, Y., Zhou, Y.: The anti-symmetric orthogonal anti-symmetric solution of a linear matrix equation and its optimal approximation. J. Hunan Univ. 2, 106–110 (2004)
Peng, Y.X., Hu, X.Y., Zhang, L.: The symmetric ortho-symmetric solution of linear matrix equation A T XA=B and its optimal approximation. Numer. Math. J. Chinese Univ. 4, 372–377 (2003)
Dulov, E.V.: Algorithms for solving matrix polynomial equations of special form. J. Appl. Math. Comput. 7, 41–60 (2000). (old KJCAM)
Zhang, X.: The general hermitian nonnegative-definite and positive-definite solutions to the matrix equation GXG *+HYH *=C. J. Appl. Math. Comput. 14, 51–67 (2004)
Paige, C.C., Saunders, M.A.: Towards a generalized singular value decomposition. SIAM J. Numer. Anal. 18, 398–405 (1981)
Stewart, G.W.: Computing the CS-decomposition of a partitioned orthogonal matrix. Numer. Math. 40, 297–306 (1982)
Golub, C.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins University Press, Baltimore (1983)
Cheney, E.W.: Introduction to Approximation Theory. McGraw–Hill, New York (1966)
Zhang, L.: The approximation on the closed convex cone and its numerical application. Hunan Ann. Math. 6, 43–48 (1986)
Peng, Z.Y., Hu, X.Y., Zhang, L.: One kind of inverse problems for symmetric and skew anti-symmetric matrices. Numer. Math. J. Chinese Univ. 2, 144–152 (2003)