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Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications. (English) Zbl 1149.15012
Maximal and minimal ranks of the quaternion matrix $$C_4 - A_4XB_4$$ are studied, where $$X$$ is a variable quaternion matrix subject to the quaternion equations $$A_1X = C_1$$, $$XB_2 = C_2$$, $$A_3XB_3 = C_3$$. As a corollary necessary and sufficient conditions for solvability of a system of quaternion matrix equations are obtained. Extremal ranks of the generalized Schur complement of a certain matrix (under some linear constraints) are also studied. The paper generalizes some previous results in this area.

##### MSC:
 15A24 Matrix equations and identities 15B33 Matrices over special rings (quaternions, finite fields, etc.) 15A09 Theory of matrix inversion and generalized inverses
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##### References:
  Hungerford, T.W., Algebra, (1980), Spring-Verlag New York Inc. · Zbl 0442.00002  Mitra, S.K., The matrix equations $$\mathit{AX} = C$$, $$\mathit{XB} = D$$, Linear algebra appl., 59, 171-181, (1984)  Bhimasankaram, P., Common solutions to the linear matrix equations $$\mathit{AX} = B, \mathit{XC} = D$$, and $$\mathit{EXF} = G$$, Sankhya ser. A, 38, 404-409, (1976) · Zbl 0411.15008  Lin, C.Y.; Wang, Q.W., New solvability conditions and a new expression of the general solution to a system of linear matrix equations over an arbitrary division ring, Southeast Asian bull. math., 29, 5, 755-762, (2005) · Zbl 1087.15018  Lin, C.Y.; Wang, Q.W., The minimal and maximal ranks of the general solution to a system of matrix equations over an arbitrary division ring, Math. sci. res. J., 10, 3, 57-65, (2006) · Zbl 1142.15302  Wang, Q.W., A system of four matrix equations over von Neumann regular rings and its applications, Acta math. sin., 21, 2, 323-334, (2005) · Zbl 1083.15021  Tian, Y.; Cheng, S., The maximal and minimal ranks of $$A - \mathit{BXC}$$ with applications, New York J. math., 9, 345-362, (2003) · Zbl 1036.15004  Tian, Y., Upper and lower bounds for ranks of matrix expressions using generalized inverses, Linear algebra appl., 355, 187-214, (2002) · Zbl 1016.15003  Ando, T., Generalized Schur complements, Linear algebra appl., 27, 173-186, (1979) · Zbl 0412.15006  Fiedler, M., Remarks on the Schur complement, Linear algebra appl., 39, 189-195, (1981) · Zbl 0465.15004  Carlson, D.; Haynsworth, E.; Markham, T., A generalization of the Schur complements by means of the Moore-Penrose inverse, SIAM J. appl. math., 26, 169-179, (1974) · Zbl 0245.15002  Carlson, D., What are Schur complements, anyway?, Linear algebra appl., 74, 257-275, (1986) · Zbl 0595.15006  Tian, Y., More on maximal and minimal ranks of Schur complements with applications, Appl. math. comput., 152, 3, 675-692, (2004) · Zbl 1077.15005  Adler, S.L., Quaternionic quantum mechanics and quantum fields, (1995), Oxford University Press Oxford · Zbl 0885.00019  Zhang, F., Quaternions and matrices of quaternions, Linear algebra appl., 251, 21-57, (1997) · Zbl 0873.15008  Baker, A., Right eigenvalues for quaternionic matrices: a topological approach, Linear algebra appl., 286, 303-309, (1999) · Zbl 0941.15013  Merino, D.I.; Sergeichuk, V., Littlewood’s algorithm and quaternion matrices, Linear algebra appl., 298, 193-208, (1999) · Zbl 0984.15017  De Leo, S.; Scolarici, G., Right eigenvalue equation in quaternionic quantum mechanics, J. phys. A, 33, 2971-2995, (2000) · Zbl 0954.81008  Farenick, D.R.; Pidkowich, B.A.F., The spectral theorem in quaternions, Linear algebra appl., 371, 75-102, (2003) · Zbl 1030.15015  Moxey, C.E.; Sangwine, S.J.; Ell, T.A., Hypercomplex correlation techniques for vector images, IEEE trans. signal process., 51, 7, 1941-1953, (2003) · Zbl 1369.94032  Le Bihan, N.; Mars, J., Singular value decomposition of matrices of quaternions: a new tool for vector-sensor signal processing, Signal process., 84, 7, 1177-1199, (2004) · Zbl 1154.94331  N. Le Bihan, S.J. Sangwine, Quaternion principal component analysis of color images, in: IEEE International Conference on Image Processing (ICIP), Barcelona, Spain, September, 2003.  N. Le Bihan, S.J. Sangwine, Color image decomposition using quaternion singular value decomposition, in: IEEE International Conference on Visual Information Engineering (VIE), Guildford, UK, July, 2003. · Zbl 1109.65037  Sangwine, S.J.; Le Bihan, N., Quaternion singular value decomposition based on bidiagonalization to a real or complex matrix using quaternion Householder transformations, Appl. math. comput., 182, 1, 727-738, (2006) · Zbl 1109.65037  Wang, Q.W., The general solution to a system of real quaternion matrix equations, Comput. math. appl., 49, 665-675, (2005) · Zbl 1138.15004  Wang, Q.W., Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. math. appl., 49, 641-650, (2005) · Zbl 1138.15003  Wang, Q.W.; Song, G.J.; Lin, C.Y., 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  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  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  Mitra, S.K., A pair of simultaneous linear matrix equations $$A_1 \mathit{XB}_1 = C_1$$ and $$A_2 \mathit{XB}_2 = C_2$$ and a programming problem, Linear algebra appl., 131, 107-123, (1990)  Wang, Q.W.; Qin, F.; Lin, C.Y., The common solution to matrix equations over a regular ring with applications, Indian J. pure appl. math., 36, 12, 655-672, (2005) · Zbl 1104.15014
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