×

Minimal ranks of some quaternion matrix expressions with applications. (English) Zbl 1204.15005

Let \(p(X, Y) = A - BX - X^{(*)}B^{(*)} - CYC^{(*)}\) and \(q(X, Y) = A - BX + X^{(*)}B^{(*)} - CYC^{(*)}\) be quaternion matrix expressions, where \(A\) is persymmetric or perskew-symmetric. The authors derive the minimal rank formula of \(p(X, Y)\) with respect to pair of matrices \(X\) and \(Y = Y^{(*)}\), and the minimal rank formula of \(q(X, Y)\) with respect to pair of matrices \(X\) and \(Y = - Y^{(*)}\). As applications, they establish some necessary and sufficient conditions for the existence of the general (persymmetric or perskew-symmetric) solutions to some well-known linear quaternion matrix equations. The expressions are also given for the corresponding general solutions of the matrix equations when the solvability conditions are satisfied. At the same time, some useful consequences are also developed.

MSC:

15A03 Vector spaces, linear dependence, rank, lineability
15B33 Matrices over special rings (quaternions, finite fields, etc.)
15A24 Matrix equations and identities
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Hungerford, T. W., Algebra (1980), Spring-Verlag Inc.: Spring-Verlag Inc. New York · Zbl 0442.00002
[2] Hamilton, W. R., On quaternions, or on a new system of imaginaries in algebra, Philosophical Magazine, 25, 3, 489-495 (1844)
[3] Conway, J. H.; Smith, D. A., On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry (2002), A K Peters: A K Peters Natick
[4] Kamberov, G.; Norman, P.; Pedit, F.; Pinkall, U., Quaternions Spinors and Surfaces. Quaternions Spinors and Surfaces, Contemporary Mathematics, vol. 299 (2002), Amer. Math. Soc.: Amer. Math. Soc. Province · Zbl 1022.53001
[5] Nebe, G., Finite quaternionic matrix groups, Represent. Theory, 2, 106-223 (1998) · Zbl 0901.20035
[6] Ward, J. P., Quaternions and Cayley Numbers. Quaternions and Cayley Numbers, Mathematics and Its Applications, vol. 403 (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht, the Netherlands · Zbl 0877.15031
[7] Shoemake, K., Animating rotation with quaternion curves, Comput. Graph., 19, 3, 245-254 (1985)
[8] Kuipers, J. B., Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality (2002), Princeton Univ. Press: Princeton Univ. Press Princeton · Zbl 1053.70001
[9] Adler, S. L., Quaternionic Quantum Mechanics and Quantum Fields (1995), Oxford University Press: Oxford University Press New York · Zbl 0885.00019
[11] Zhang, F., Quaternions and matrices of quaternions, Linear Algebra Appl., 251, 21-57 (1997) · Zbl 0873.15008
[12] 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
[13] Reid, R. M., Some eigenvalues properties of persymmetric matrices, SIAM Rev., 39, 313-316 (1997) · Zbl 0876.15006
[14] Eberle, M. G.; Maciel, M. C., Finding the closest persymmetric and skew-symmetric matrices, Mecánica Comput., XXII, 1789-1802 (2003)
[15] Mitra, S. K., Fixed rank solutions of linear matrix equations, Sankhya Ser. A, 35, 387-392 (1972) · Zbl 0261.15008
[16] Mitra, S. K., The matrix equations AX=C, XB=D, Linear Algebra Appl., 59, 171-181 (1984) · Zbl 0543.15011
[17] 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
[18] Mitra, S. K., A pair of simultaneous linear matrix equations \(A_1 XB_1=C_1, A_2 XB_2=C_2\) and a programming problem, Linear Algebra Appl., 131, 107-123 (1990)
[19] Tian, Y., Ranks of solutions of the matrix equation \(AXB =C\), Linear Multilinear Algebra, 51, 2, 111-125 (2003) · Zbl 1040.15003
[20] Tian, Y.; Cheng, S., The maximal and minimal ranks of \(A\)−BXC with applications, New York J. Math., 9, 45-362 (2003) · Zbl 1036.15004
[21] Wang, Q. W.; Wu, Z. C.; Lin, C. Y., Extremal ranks of a quaternion matrix expression subject to consistent systems of quaternion matrix equations with applications, Appl. Math. Comput., 182, 1755-1764 (2006) · Zbl 1108.15014
[22] Tian, Y., Completing block matrices with maximal and minimal ranks, Linear Algebra Appl., 321, 327-345 (2000) · Zbl 0984.15013
[23] Tian, Y., The minimal rank of the matrix expression \(A\)−BXYC, Missouri J. Math. Sci., 14, 1, 40-48 (2002) · Zbl 1032.15001
[24] Tian, Y., The minimal rank completion of a 3×3 partial block matrix, Linear Multilinear Algebra, 50, 2, 125-131 (2002) · Zbl 1006.15004
[25] Tian, Y., Upper and lower bounds for ranks of matrix expressions using generalized inverses, Linear Algebra Appl., 355, 187-214 (2002) · Zbl 1016.15003
[26] Tian, Y., The maximal and minimal ranks of some expressions of generalized inverses of matrices, Southeast Asian Bull. Math., 25, 745-755 (2002) · Zbl 1007.15005
[27] Tian, Y., More on maximal and minimal ranks of Schur complements with applications, Appl. Math. Comput., 152, 3, 675-692 (2004) · Zbl 1077.15005
[28] Tian, Y.; Liu, Y., Extremal ranks of some symmetric matrix expressions with applications, SIAM J. Matrix Anal. A, 28, 890-905 (2006) · Zbl 1123.15001
[29] Marsaglia, G.; Styan, G. P.H., Equalities and inequalities for ranks of matrices, Linear Multilinear Algebra, 2, 269-292 (1974) · Zbl 0297.15003
[30] Hua, D., On the symmetric solutions of linear matrix equations, Linear Algebra Appl., 131, 1-7 (1990) · Zbl 0712.15009
[31] Hua, D.; Lancaster, P., Linear matrix equations from an inverse problem of vibration theory, Linear Algebra Appl., 246, 31-47 (1996) · Zbl 0861.15014
[32] Wang, Q. W., A system of matrix equations and a linear matrix equation over arbitrary regular rings with identity, Linear Algebra Appl., 384, 43-54 (2004) · Zbl 1058.15015
[33] Wang, Q. W.; Zhou, Y.; Zhang, Q., Ranks of the common solution to six quaternion matrix equations, Acta Math. Appl. Sinica (English Series) (2010)
[34] Wang, Q. W.; Yu, S. W., Extreme ranks of real matrices in solution of the quaternion matrix equation \(AXB =C\) with applications, Algebr. Colloq., 17, 2, 345-360 (2010) · Zbl 1188.15016
[35] Wang, Q. W.; van der Woude, J. W.; Chang, H. X., A system of real quaternion matrix equations with applications, Linear Algebra Appl., 431, 2291-2303 (2009) · Zbl 1180.15019
[36] 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
[37] 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
[38] Wang, Q. W.; Zhang, H. S.; Song, G. J., A new solvable condition for a pair of generalized Sylvester equations, Electronical J. Linear Algebra, 18, 289-301 (2009) · Zbl 1190.15019
[39] 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, Algebr. Colloq., 16, 2, 293-308 (2009) · Zbl 1176.15020
[40] Wang, Q. W.; Zhang, H. S.; Yu, S. W., On the real and pure imaginary solutions to the quaternion matrix equation \(AXB + CYD =E\), Electronical J. Linear Algebra, 17, 343-358 (2008) · Zbl 1154.15019
[41] 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
[42] Duan, X.; Liao, A., On the existence of Hermitian positive definite solutions of the matrix equation \(X^s+A^∗X^{−t}A=Q\), Linear Algebra Appl., 429, 4, 673-687 (2008) · Zbl 1143.15011
[43] Wang, Q. W., The general solution to a system of real quaternion matrix equations, Comput. Math. Appl., 49, 665-675 (2005) · Zbl 1138.15004
[44] Wang, Q. W., Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. Math. Appl., 49, 641-650 (2005) · Zbl 1138.15003
[45] Tian, Y., The solvability of two linear matrix equations, Linear Multilinear Algebra, 48, 123-147 (2000) · Zbl 0970.15005
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.