The common bisymmetric nonnegative definite solutions with extreme ranks and inertias to a pair of matrix equations. (English) Zbl 1267.15017
The paper considers the bisymmetric nonnegative definite solution with extremal ranks and inertias to a system of quaternion matrix equations \(AX=C\), \(XB=D\). First, the authors give a general expression of bisymmetric nonnegative definite solutions to the system when the solvablility conditions are satisfied. Then they drive the representation of these solutions with extremal ranks and inertias, and finally, they give a numerical example to illustrate their results.
Reviewer: Yohannes Tadesse (Uppsala)
MSC:
| 15A24 | Matrix equations and identities |
| 15B33 | Matrices over special rings (quaternions, finite fields, etc.) |
Keywords:
system of quaternion matrix equations; rank; inertia; bisymmetric matrix; bisymmetric nonnegative definite matrix; numerical exampleReferences:
| [1] | Liao, A. P.; Bai, Z. Z., The constrained solutions of two matrix equations, Acta Math. Sinica Engl. Ser., 18, 4, 671-678 (2002) · Zbl 1028.15011 |
| [2] | Liao, A. P.; Bai, Z. Z., Least-squares solution of \(AXB =D\) over symmetric positive semidefinite matrices, J. Comput. Math., 21, 175-182 (2003) · Zbl 1029.65042 |
| [3] | 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 |
| [4] | Xie, D. X.; Zhang, L.; Hu, X. Y., The solvability conditions and the inverse problem of bisymmetric nonnegative definite matrices, J. Comput. Math., 6, 597-608 (2000) · Zbl 0966.15008 |
| [5] | Henk Don, F. J., On the symmetric solutions of a linear matrix equation, Linear Algebra Appl., 93, 1-7 (1987) · Zbl 0622.15001 |
| [6] | Dai, H., On the symmetric solutions of linear matrix equations, Linear Algebra Appl., 131, 1-7 (1990) · Zbl 0712.15009 |
| [7] | Marsaglia, G.; Styan, G. P.H., Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra, 2, 269-292 (1974) · Zbl 0297.15003 |
| [8] | Zhao, L. J.; Hu, X. Y.; Zhang, L., Least squares solutions to \(AX =B\) for bisymmetric matrices under a central principal submatrix constraint and the optimal approximation, Linear Algebra Appl., 428, 871-880 (2008) · Zbl 1133.15016 |
| [9] | Dehghan, M.; Hajarian, M., An iterative method for solving the generalized coupled Sylvester matrix equations over generalized bisymmetric matrices, Appl. Math. Model., 34, 639-654 (2010) · Zbl 1185.65054 |
| [10] | Dehghan, M.; Hajarian, M., The general coupled matrix equations over generalized bisymmetric matrices, Linear Algebra Appl., 432, 6, 1531-1552 (2010) · Zbl 1187.65042 |
| [11] | Uhlig, F., On the matrix equation \(AX =B\) with applications to the generators of controllability matrix, Linear Algebra Appl., 85, 203-209 (1987) · Zbl 0612.15006 |
| [12] | Wang, Q. W., Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. Math. Appl., 49, 641-650 (2005) · Zbl 1138.15003 |
| [13] | 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 |
| [14] | Zhang, Q.; Wang, Q. W., The \((P,Q)\)-(skew)symmetric extremal rank solutions to a system of quaternion matrix equations, Appl. Math. Comput., 217, 9286-9296 (2011) · Zbl 1222.15022 |
| [15] | Wang, Q. W.; Chang, H. X.; Lin, C. Y., P-(skew)symmetric common solutions to a pair of quaternion matrix equations, Appl. Math. Comput., 195, 721-732 (2008) · Zbl 1149.15011 |
| [16] | Xiao, Q. F.; Hu, X. Y.; Zhang, L., The symmetric minimal rank solution of the matrix equation \(AX =B\) and the optimal approximation, Electron. J. Linear Algebra, 18, 264-271 (2009) · Zbl 1171.65387 |
| [17] | Mitra, S. K., A pair of simultaneous linear matrix equations \(A_1 XB_1=C_1, A_2 XB_2=C_2\) and a matrix programming problem, Linear Algebra Appl., 131, 107-123 (1990) · Zbl 0712.15010 |
| [18] | Mitra, S. K., The matrix equations \(AX =C, XB =D\), Linear Algebra Appl., 59, 171-182 (1984) · Zbl 0543.15011 |
| [19] | Wang, Q. W.; Zhou, Y.; Zhang, Q., Ranks of the common solution to six quaternion matrix equations, Acta Math. Applic. Sinica (Engl. Ser.), 27, 3, 443-462 (2011) · Zbl 1304.15006 |
| [20] | Lian, D. Z.; Wang, Q. W.; Tang, Y., Extreme ranks of a partial banded block quaternion matrix expression subject to some matrix equations with applications, Algebra Colloq., 18, 2, 333-346 (2011) · Zbl 1227.15015 |
| [21] | Wang, Q. W.; Jiang, Jing, Extreme ranks of (skew-)Hermitian solutions to a quaternion matrix equation, Electron. J. Linear Algebra, 20, 552-573 (2010) · Zbl 1207.15016 |
| [22] | Khan, Israr Ali; Wang, Q. W.; Song, G. J., Minimal ranks of some quaternion matrix expressions with applications, Appl. Math. Comput., 217, 2031-2040 (2010) · Zbl 1204.15005 |
| [23] | Wang, Q. W.; Yu, S. W., Extreme ranks of real matrices in solution of the quaternion matrix equation \(AXB =C\) with applications, Algebra Colloq., 17, 2, 345-360 (2010) · Zbl 1188.15016 |
| [24] | Wang, Q. W.; Zhang, H. S.; Song, G. J., A new solvable condition for a pair of generalized Sylvester equations, Electron. J. Linear Algebra, 18, 289-301 (2009) · Zbl 1190.15019 |
| [25] | Wang, Q. W.; Yu, S. W., The real solution to a system of quaternion matrix equations with applications, Commun. Algebra, 37, 6, 2060-2079 (2009) · Zbl 1393.15021 |
| [26] | Wang, Q. W.; Li, C. K., Ranks and the least-norm of the general solution to a system of quaternion matrix equations, Linear Algebra Appl., 430, 1626-1640 (2009) · Zbl 1158.15010 |
| [27] | Wang, Q. W.; Zhang, H. S.; Yu, S. W., On the real and pure imaginary solutions to the quaternion matrix equation \(AXB + CYD =E\), Electron. J. Linear Algebra, 17, 343-358 (2008) · Zbl 1154.15019 |
| [28] | Wang, Q. W.; Song, G. J., Maximal and minimal ranks of the common solution of some linear matrix equations over an arbitrary division ring with applications, Algebra Colloq., 16, 2, 293-308 (2009) · Zbl 1176.15020 |
| [29] | Wang, Q. W.; Song, G. J.; Lin, C. Y., Rank equalities related to the generalized inverse with applications, Appl. Math. Comput., 205, 370-382 (2008) · Zbl 1159.15006 |
| [30] | Wang, Q. W.; Song, G. J., Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application, Appl. Math. Comput., 189, 1517-1532 (2007) · Zbl 1124.15010 |
| [31] | Wang, Q. W.; Wu, Z. C.; Lin, C. Y., Extremal ranks of a quaternion matrix expression subject to consistent system of quaternion matrix equations with applications, Appl. Math. Comput., 182, 1755-1764 (2006) · Zbl 1108.15014 |
| [32] | Hu, X. Y.; Zhang, L.; Xie, D. X., The solvability conditions for the inverse eigenvalue problem of bisymmetric matrices, Math. Numer. Sinica, 4, 409-418 (1998) · Zbl 0923.15006 |
| [33] | Chu, D. L., Inertia and rank characterizations of some expressions, SIAM J. Matrix Anal. Appl., 3, 1187-1226 (2009) · Zbl 1198.15010 |
| [34] | Tian, Y., Equalities and inequalities for inertias of Hermitian matrices with applications, Linear Algebra Appl., 433, 263-296 (2010) · Zbl 1205.15033 |
| [35] | Deng, Y. B.; Hu, X. Y.; Zhang, L., Least-squares solution of \(BXA^T=T\) over symmetric, skew-symmetric, and positive semidefinite \(X\), SIAM J. Matrix Anal. Appl., 25, 486-494 (2003) · Zbl 1050.65037 |
| [36] | Tian, Y.; Liu, Y. H., Extremal ranks of some symmetric matrix expressions with applications, SIAM J. Matrix Anal. Appl., 28, 890-905 (2006) · Zbl 1123.15001 |
| [37] | Xie, D.; Hu, X.; Sheng, Y. P., The solvability conditions for the inverse eigenproblems of symmetric and generalized centro-symmetric matrices and their approximations, Linear Algebra Appl., 418, 1, 142-152 (2006) · Zbl 1109.65034 |
| [38] | Xie, D.; Zhang, Z.; Liu, Z., Theory and method for updating least-squares finite element model of symmetric generalized centro-symmetric matrices, J. Comput. Appl. Math., 216, 2, 484-497 (2008) · Zbl 1138.74050 |
| [39] | Yuan, S.; Liao, A.; Lei, Y., Inverse eigenvalue problems of tridiagonal symmetric matrices and tridiagonal bisymmetric matrices, Comput. Math. Applic., 55, 11, 2521-2532 (2008) · Zbl 1142.65318 |
| [40] | Yuan, S.; Liao, A.; Lei, Y., Least squares Hermitian solution of the matrix equation (AXB,CXD)=\((E,F)\) with the least norm over the skew field of quaternions, Math. Comput. Model., 48, 1-2, 91-100 (2008) · Zbl 1145.15303 |
| [41] | Peng, Z. H.; Hu, X. Y.; Zhang, L., The bisymmetric solutions of the matrix equation \(A_1X_1B_1+A_2X_2B_2\)+⋯+\(A_lX_{l\) · Zbl 1123.15009 |
| [42] | Wang, Q. W.; Zhang, F., The reflexive re-nonnegative definite solution to a quaternion matrix equation, Electron. J. Linear Algebra, 17, 88-101 (2008) · Zbl 1147.15012 |
| [43] | Wang, Q. W.; Chang, H. X., On the centro-symmetric solution of a system of matrix equations over a regular ring with identity, Algebra Colloq., 14, 4, 555-570 (2007) · Zbl 1143.15013 |
| [44] | Peng, Z. Y.; Hu, X. Y.; Zhang, L., The inverse problem of bisymmetric matrices with a submatrix constraint, Numer. Linear Algebra Appl., 11, 59-73 (2004) · Zbl 1164.15322 |
| [45] | Wang, Q. W.; van der Woude, J. W.; Yu, S. W., An equivalence canonical form of a matrix triplet over an arbitrary division ring with applications, Sci. China Math., 54, 5, 907-924 (2011) · Zbl 1218.15008 |
| [46] | Wang, Q. W.; Dong, C. Z., Positive solutions to a system of adjointable operator equations over Hilbert \(C^∗\)-modules, Linear Algebra Appl., 433, 1481-1489 (2010) · Zbl 1214.47015 |
| [47] | Wang, Q. W.; Wu, Z. C., Common Hermitian solutions to some operator equations on Hilbert \(C^∗\)-modules, Linear Algebra Appl., 432, 2, 3159-3171 (2010) · Zbl 1197.47031 |
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