×

zbMATH — the first resource for mathematics

Least-squares solution with the minimum-norm for the matrix equation \((A\times B,G\times H) = (C,D)\). (English) Zbl 1087.65040
The authors derive an analytical expression of the minimum-norm least-squares solution of the matrix equation \((A\times B, G\times H)=(C, D)\). Their approach uses the generalized singular value and canonical correlation decompositions. They verify the obtained expressions by some numerical experiments involving Toeplitz and Hankel matrices.

MSC:
65F20 Numerical solutions to overdetermined systems, pseudoinverses
15A24 Matrix equations and identities
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Kolka, G.K.G., Linear matrix equations and pole assignment, () · Zbl 0592.93023
[2] Woude, J.V., Feedback decoupling and stabilization for linear system with multiple exogenous variables, ()
[3] Mitra, S.K., Common solutions to a pair of linear matrix equations A1XB1 = C1 and A2XB2 = C2, (), 213-216
[4] Chu, K.-E., Singular value and generalized singular value decomposition and the solution of linear matrix equation, Linear algebra appl., 87, 83-98, (1987) · Zbl 0612.15003
[5] Liao, A.P., A generalization of a class of inverse eigenvalue problem, J. hunan univ., 22, 7-10, (1995) · Zbl 0832.65038
[6] Mitra, S.K., A pair of simultaneous linear matrix equations A1XB1 = C1 and A2XB2 = C2 and a matrix programming problem, Linear algebra appl., 131, 107-123, (1990)
[7] Navarra, A.; Odell, P.L.; Young, D.M., A representation of the general common solution to the matrix equations A1XB1 = C1 and A2XB2 = C2 with applications, Computers math. applic., 41, 7/8, 929-935, (2001) · Zbl 0983.15016
[8] Woude, J.V., On the existence of a common solution X to the matrix equations aixbj = cij, (i, j) ɛ γ, Linear algebra appl., 375, 135-145, (2003)
[9] Yuan, Y.X., On the two classes of best approximation problems, Math. numerica sinica, 23, 429-436, (2001)
[10] Yuan, Y.X., The minimum norm solutions of two classes of matrix equations, Numer. math.-A J. Chinese univ., 24, 127-134, (2002) · Zbl 1021.15010
[11] Yuan, Y.X., The optimal solution of linear matrix equation by matrix decompositions, Math. numerica sinica, 24, 165-176, (2002)
[12] Shinozaki, N.; Sibuya, M., Consistency of a pair of matrix equations with an application, Keio engrg. rep., 27, 141-146, (1974)
[13] Roth, W.E., The equations AX − YB = C and AX − XB = C in matrices, (), 392-396 · Zbl 0047.01901
[14] Wang, R.S., Functional analysis and optimization theory, (2003), Beijing Univ. of Aeronautics & Astronautics Press Netherlands
[15] Paige, C.C.; Saunders, M.A., Towards a generalized singular value decomposition, SIAM J. numer. anal., 18, 398-405, (1981) · Zbl 0471.65018
[16] Stewart, G.W., Computing the CS-decomposition of a partitioned orthogonal matrix, Numer. math., 40, 298-306, (1982) · Zbl 0516.65016
[17] Golub, G.H.; Zha, H., Perturbation analysis of the canonical correlations of matrix pairs, Linear algebra appl., 210, 3-28, (1994) · Zbl 0811.15011
[18] Golub, G.H.; Van Loan, C.F., Matrix computations, (1997), The Johns Hopkins Univ. Press Beijing
[19] Stewart, G.W.; Sun, J.G., Matrix perturbation theory, (1990), Academic Press Baltimore, MD
[20] Aubin, J.P., Applied functional analysis, (1979), John Wiley & Sons New York
[21] Shim, S.Y.; Chen, Y., Least squares solutions of matrix equation AXB* + CYD* = E, SIAM J. matrix anal. appl., 24, 802-808, (2003) · Zbl 1037.65042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.