## 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
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### References:

 [1] Kolka, G. K.G., Linear matrix equations and pole assignment, (Ph.D. Thesis (1984), Dept. of Mathematics, and Computer Science, Univ. of Salford: Dept. of Mathematics, and Computer Science, Univ. of Salford Cambridge) · Zbl 0592.93023 [2] Woude, J. V., Feedback decoupling and stabilization for linear system with multiple exogenous variables, (Ph.D. Thesis (1987), Technical Univ. of Eindhoven: Technical Univ. of Eindhoven U.K.) [3] Mitra, S. K., Common solutions to a pair of linear matrix equations $$A_1 XB_1 = C_1$$ and $$A_2 XB_2 = C_2$$, (Proc. Cambridge Philos. Soc., 74 (1973)), 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 $$A_1 XB_1 = C_1$$ and $$A_2 XB_2 = C_2$$ 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 $$A_1 XB_1 = C_1$$ and $$A_2 XB_2 = C_2$$ 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 $$A_i XB_j$$ = $$C_{ij}$$, (i, j) ɛ Γ, Linear Algebra Appl., 375, 135-145 (2003) · Zbl 1037.15014 [9] Yuan, Y. X., On the two classes of best approximation problems, Math. Numerica Sinica, 23, 429-436 (2001) · Zbl 1495.15022 [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) · Zbl 1495.15026 [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, (Proc. Amer. Math. Soc., 3 (1952)), 392-396 · Zbl 0047.01901 [14] Wang, R. S., Functional Analysis and Optimization Theory (2003), Beijing Univ. of Aeronautics & Astronautics Press: 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: The Johns Hopkins Univ. Press Beijing [19] Stewart, G. W.; Sun, J. G., Matrix Perturbation Theory (1990), Academic Press: Academic Press Baltimore, MD [20] Aubin, J. P., Applied Functional Analysis (1979), John Wiley & Sons: 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
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