Liu, Yong Hui Ranks of solutions of the linear matrix equation \(AX + YB = C\). (English) Zbl 1129.15009 Comput. Math. Appl. 52, No. 6-7, 861-872 (2006). The author investigates the maximal and minimal ranks of a solution \((X,Y)\) to the complex matrix equation \(AX+YB=C\). Moreover, the maximal and minimal ranks of four real matrices \(X_0,X_1,Y_0,Y_1\) in a pair of solutions \(X=X_0+iX_1\) and \(Y=Y_0+iY_1\) to the matrix equation mentioned above are also presented. Reviewer: Qing-Wen Wang (Shanghai) Cited in 1 ReviewCited in 19 Documents MSC: 15A24 Matrix equations and identities 15A09 Theory of matrix inversion and generalized inverses Keywords:maximal rank; minimal rank; generalized inverse PDFBibTeX XMLCite \textit{Y. H. Liu}, Comput. Math. Appl. 52, No. 6--7, 861--872 (2006; Zbl 1129.15009) Full Text: DOI References: [1] Baksalary, J. K.; Kala, P., The matrix equation \(AX + YB =C\), Linear Algebra Appl., 25, 41-43 (1979) · Zbl 0403.15010 [2] Chu, K. E., The solution of the matrix equations \(AXB − CXD =E\) and \((YA − DZ, YC − BZ) =\)(E, F), Linear Algebra Appl., 93, 93-105 (1987) · Zbl 0631.15006 [3] Franders, H.; Wimmer, H., On the matrix equations \(AX − XB =C\) and \(AX − YB =C\), SIAM J. Appl. Math., 32, 707-710 (1977) · Zbl 0385.15008 [4] Hartwig, R., Roth equivalence problem in unit regular rings, Proc. Amer. Math. Soc., 59, 39-44 (1976) · Zbl 0347.15005 [5] Roth, R., The equations \(AX − YB =C\) and AX−XB=c in matrices, Proc. Amer. Math. Soc., 3, 392-396 (1952) · Zbl 0047.01901 [6] Tian, Y., The minimal rank of the matrix expression \(A\)−BX−YC, Missouri J. Math. Sci., 14, 40-48 (2002) · Zbl 1032.15001 [7] Tian, Y., Upper and lower bounds for ranks of matrix expressions using generalized inverses, Linear Algebra Appl., 355, 187-214 (2002) · Zbl 1016.15003 [8] Tian, Y., Ranks of solutions of the matrix equation \(AXB =C\), Linear and Multilinear Algebra, 51, 111-125 (2003) · Zbl 1040.15003 [9] Marsaglia, G.; Styan, G. P.H., Equalities and inequalities for ranks of matrices, Linear and Multilinear Algebra, 2, 269-292 (1974) · Zbl 0297.15003 [10] 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 [11] Ben-Israel, A.; Greville, T. N.E., Generalized Inverses: Theory and Applications (2003), Springer-Verlag: Springer-Verlag New York · Zbl 1026.15004 [12] Rao, C. R.; Mitra, S. K., Generalized Inverse of Matrices and Its Applications (1971), Wiley: Wiley New York 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.