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A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity. (English) Zbl 1058.15015
Many problems in systems and control theory require the solution of Sylvester’s equation $AX-YB=C$ or of its generalization $(*)$ $AXB+CYD=E$. The author studies the couple of matrix equations $(**)$ $A_1XB_1=C_1,A_2XB_2=C_2$ over an arbitrary regular ring with identity. He obtains necessary and sufficient conditions for the consistency of the system $(**)$ and presents its general solution. The results are used to obtain necessary and sufficient conditions for the consistency of the equation $(*)$ and to derive the form of its general solution.

##### MSC:
 15A24 Matrix equations and identities 15A06 Linear equations (linear algebra) 15B33 Matrices over special rings (quaternions, finite fields, etc.) 16E50 Von Neumann regular rings and generalizations 15A09 Matrix inversion, generalized inverses
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##### References:
 [1] Mitra, S. K.: A pair of simultaneous linear matrix equations A1XB1=C1 and A2XB2=C2. Proc. Cambridge philos. Soc. 74, 213-216 (1973) [2] Mitra, S. K.: A pair of simultaneous linear matrix equations and a matrix programming problem. Linear algebra appl. 131, 97-123 (1990) · Zbl 0712.15010 [3] Shinozaki, N.; Sibuya, M.: Consistency of a pair of matrix equations with an application. Keio eng. Rep. 27, 141-146 (1974) · Zbl 0409.15010 [4] J. Van der Woulde, Feedback decoupling and stabilization for linear system with multiple exogenous variables, Ph.D. Thesis, Technical University of Eindhoven, Netherlands, 1987 [5] Van Der Woulde, J.: Almost noninteracting control by measurement feedback. Systems control lett. 9, 7-16 (1987) · Zbl 0623.93028 [6] Özgüler, A. B.; Akar, N.: A common solution to a pair of linear matrix equations over a principle domain. Linear algebra appl. 144, 85-99 (1991) · Zbl 0718.15006 [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. Comput. math. Appl. 41, 929-935 (2001) · Zbl 0983.15016 [8] Wang, Q. W.: The decomposition of pairwise matrices and matrix equations over an arbitrary skew field. Acta math. Sinica 39, No. 3, 396-403 (1996) · Zbl 0870.15007 [9] Roth, W. E.: The equation AX-YB=C and AX-XB=C in matrices. Proc. amer. Math. soc. 3, 392-396 (1952) · Zbl 0047.01901 [10] Hartwig, R. E.: Roth’s equivalence problem in unit regular rings. Proc. amer. Math. soc. 59, 39-44 (1976) · Zbl 0347.15005 [11] Hartwig, R.: Roth’s removal rule revisited. Linear algebra appl. 49, 91-115 (1984) · Zbl 0509.15004 [12] Huylebrouck, D.: The generalized inverses of a sum with radical elements: applications. Linear algebra appl. 246, 159-175 (1996) · Zbl 0862.15004 [13] Guralnick, R. M.: Matrix equivalence and isomorphism of modules. Linear algebra appl. 43, 125-136 (1982) · Zbl 0493.16015 [14] Guralnick, R. M.: Roth’s theorems and decomposition of modules. Linear algebra appl. 39, 155-165 (1981) · Zbl 0468.16022 [15] Guralnick, R. M.: Roth’s theorems for sets of matrices. Linear algebra appl. 71, 113-117 (1985) · Zbl 0584.15006 [16] Gustafson, W.; Zelmanowitz, J.: On matrix equivalence and matrix equations. Linear algebra appl. 27, 219-224 (1979) · Zbl 0419.15009 [17] Gustafson, W.: Roth’s theorems over commutative rings. Linear algebra appl. 23, 245-251 (1979) · Zbl 0398.15013 [18] Wimmer, H. K.: Roth’s theorems for matrix equations with symmetry constraints. Linear algebra appl. 199, 357-362 (1994) · Zbl 0796.15014 [19] Wimmer, H. K.: Linear matrix equations: the module theoretic approach. Linear algebra appl. 120, 149-164 (1989) · Zbl 0677.15001 [20] Wimmer, H. K.: Consistency of a pair of generalized Sylvester equations. IEEE trans. Automat. control 39, 1014-1016 (1994) · Zbl 0807.93011 [21] Beitia, M. A.; Gracia, J. M.: Sylvester matrix equation for matrix quadruples. Linear algebra appl. 232, 155-197 (1996) · Zbl 0840.15010 [22] Huang, L.; Liu, J.: The extension of Roth’s theorem for matrix equations over a ring. Linear algebra appl. 259, 229-235 (1997) · Zbl 0880.15016 [23] Q.W. Wang, Roth’s theorems for centroselfconjugate and centroskewselfconjugate solutions to systems of linear matrix equations over a finite dimensional central algebra, Southeast Asian Bull. Math. 27, in press · Zbl 1058.15016 [24] Wang, Q. W.; Sun, J. H.; Li, S. Z.: Consistency for $bi(skew)$ symmetric solutions to systems of generalized Sylvester equations over a finite central algebra. Linear algebra appl. 353, 169-182 (2002) · Zbl 1004.15017 [25] Wang, Q. W.: On the center (skew-)self-conjugate solutions to the systems of matrix equations over a finite dimensional central algebra. Math. sci. Res. hot-line 5, No. 12, 11-17 (2001) · Zbl 1085.15501 [26] Wang, Q. W.; Li, S. Z.: Persymmetric and perskewsymmetric solutions to sets of matrix equations over a finite central algebra. Acta math. Sinica 47, No. 1, 27-34 (2004) · Zbl 1167.15305 [27] Wang, Q. W.; Sun, J. H.: The consistency of systems of matrix equations over a finite central algebra. J. natur. Sci. math. 41, No. 1, 61-69 (2001) [28] Baksalary, J. K.; Kala, R.: The matrix equation AXB+CYD=E. Linear algebra appl. 30, 141-147 (1980) · Zbl 0437.15005 [29] Özgüler, A. B.: The matrix equation AXB+CYD=E over a principal ideal domain. SIAM J. Matrix anal. Appl. 12, 581-591 (1991) · Zbl 0742.15006 [30] Huang, L.; Zeng, Q.: The solvability of matrix equation AXB+CYD=E over a simple Artinian ring. Linear and multilinear algebra 38, 225-232 (1995) · Zbl 0824.15015 [31] Huang, L.: The solvability of matrix equation AXB+CYD=E over a ring. Adv. math. (China) 26, No. 3, 269-275 (1997) · Zbl 0886.15011 [32] Brown, B.; Mccoy, N. H.: The maximal regular ideal of a ring. Proc. amer. Math. soc. 1, 165-171 (1952) · Zbl 0036.29702