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Least squares solutions with special structure to the linear matrix equation AXB=C. (English) Zbl 1230.15010
The equation AXB=C with given matrices A, B, C plays a very important role in matrix theory and applications and has been studied extensively. It is, in particular, connected with a certain growth curve model where it is important to be able to estimate the parameter matrix X. The authors derive the maximal and minimal ranks of the submatrices of a least squares solution matrix X and from these formulas they derive necessary and sufficient conditions for the submatrices to be 0 or other special forms. Finally, they obtain necessary and sufficient conditions for a least squares solution matrix X to be Hermitian or locally Hermitian.
MSC:
15A24Matrix equations and identities
References:
[1]Pan, J. X.; Fang, K. T.: Growth curve models with statistical diagnostics, (2002)
[2]Kollo, T.; Von Rosen, D.: Advanced multivariate statistics with matrices, (2005)
[3]Mitra, S. K.: Fixed rank solutions of linear matrix equations, Sankhya ser. A. 35, 387-392 (1972) · Zbl 0261.15008
[4]Khatri, C. G.; Mitra, S. K.: Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. Appl. math. 31, 579-585 (1976) · Zbl 0359.65033 · doi:10.1137/0131050
[5]SlavĂ­k, Petr: Least extremal solution of the operator equation, J. math. Anal. appl. 148, 251-262 (1990) · Zbl 0717.47005 · doi:10.1016/0022-247X(90)90042-E
[6]Peng, Z.: An iterative method for the least squares symmetric solution of the linear matrix equation AXB=C, Appl. math. Comput. 170, 711-723 (2005) · Zbl 1081.65039 · doi:10.1016/j.amc.2004.12.032
[7]Yuan, Y.; Dai, H.: Generalized reflexive solutions of the matrix equation AXB=D and an associated optimal approximation problem, Appl. math. Comput. 56, 1643-1649 (2008) · Zbl 1155.15301 · doi:10.1016/j.camwa.2008.03.015
[8]Tian, Y.: Some properties of submatrices in a solution to the matrix equation AXB=C with applications, J. franklin I. 346, 557-569 (2009) · Zbl 1168.15307 · doi:10.1016/j.jfranklin.2009.02.013
[9]Liu, Y.: Ranks of least squares solutions of the matrix equation AXB=C, Comput. math. Appl. 55, 1270-1278 (2008) · Zbl 1157.15014 · doi:10.1016/j.camwa.2007.06.023
[10]Tian, Y.: The minimal rank of the matrix expression A - BX - YC, Missouri J. Math. sci. 14, 40-48 (2002) · Zbl 1032.15001 · doi:http://www.math-cs.cmsu.edu/~mjms/2002-1d.html
[11]Tian, Y.; Liu, Y.: Extremal ranks of some symmetric matrix expressions with applications, SIAM J. Matrix anal. Appl. 28, 890-905 (2006) · Zbl 1123.15001 · doi:10.1137/S0895479802415545
[12]Y. Tian, Rank equalities related to generalized inverses of matrices and their applications, Master Thesis, Montreal, Quebec, Canada, 2000. lt;http://arXiv.org/abs/math/0003224v1gt;.
[13]Marsaglia, G.; Styan, G. P. H.: Equalities and inequalities for ranks of matrices, Linear multilinear algebra. 2, 269-292 (1974) · Zbl 0297.15003