zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Ranks and the least-norm of the general solution to a system of quaternion matrix equations. (English) Zbl 1158.15010
The authors consider the system of linear quaternion matrix equations A 1 X 1 =C 1 , A 2 X 2 =C 2 , A 3 X 1 B 1 +A 4 X 2 B 2 =C 3 which is presumed consistent. They establish a new expression of its general solution; the system has been investigated recently by Q.-W. Wang, H.-X. Chang and C.-Y. Lin [Appl. Math. Comput. 195, No. 2, 721–732 (2008; Zbl 1149.15011)]. The authors derive the minimal and maximal ranks and the least-norm of the general solution to the system. Some previously known results are special cases of the ones in this paper.
MSC:
15A24Matrix equations and identities
15A33Matrices over special rings
15A09Matrix inversion, generalized inverses
15A03Vector spaces, linear dependence, rank
References:
[1]Hungerford, T. W.: Algebra, (1980)
[2]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 · doi:10.1016/j.amc.2007.05.021
[3]Mitra, S. K.: The matrix equations AX=C, XB=D, Linear algebra appl. 59, 171-181 (1984) · Zbl 0543.15011 · doi:10.1016/0024-3795(84)90166-6
[4]Mitra, S. K.: A pair of simultaneous linear matrix equations A1XB1=C1, A2XB2=C2 and a programming problem, Linear algebra appl. 131, 107-123 (1990) · Zbl 0712.15010 · doi:10.1016/0024-3795(90)90377-O
[5]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 · doi:10.1016/0024-3795(87)90217-5
[6]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 · doi:10.1016/j.amc.2006.06.012
[7]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 · doi:10.1016/j.amc.2006.12.039
[8]Wang, Q. W.; Yu, S. W.; Lin, C. Y.: Extreme ranks of a linear quaternion matrix expression subject to triple quaternion matrix equations with applications, Appl. math. Comput. 195, 733-744 (2008) · Zbl 1149.15012 · doi:10.1016/j.amc.2007.05.018
[9]Liu, Y. H.: Ranks of solutions of the linear matrix equation AX+YB=C, Comput. math. Appl. 52, 861-872 (2006) · Zbl 1129.15009 · doi:10.1016/j.camwa.2006.05.011
[10]Tian, Y.: Ranks of solutions of the matrix equation AXB=C, Linear and multilinear algebra 51, No. 2, 111-125 (2003) · Zbl 1040.15003 · doi:10.1080/0308108031000114631
[11]Tian, Y.: Ranks and independence of solutions of the matrix equation AXB+CYD=M, Acta math. Univ. comenianae 1, 75-84 (2006) · Zbl 1164.15321
[12]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, No. 3, 57-65 (2006) · Zbl 1142.15302
[13]Q.W. Wang, G.J. Song, Maximal and minimal ranks of the common solution of some linear matrix equations over an arbitrary division ring, Algebra Colloquium, in press. · Zbl 1176.15020
[14]Tian, Y.: Upper and lower bounds for ranks of matrix expressions using generalized inverses, Linear algebra appl. 355, 187-214 (2002) · Zbl 1016.15003 · doi:10.1016/S0024-3795(02)00345-2
[15]Tian, Y.; Cheng, S.: The maximal and minimal ranks of A-BXC with applications, New York J. Math. 9, 345-362 (2003) · Zbl 1036.15004 · doi:emis:journals/NYJM/j/2003/9-18nf.htm
[16]Tian, Y.: Solvability of two linear matrix equations, Linear and multilinear algebra 48, 123-147 (2000) · Zbl 0970.15005 · doi:10.1080/03081080008818664
[17]Wang, Q. W.; Chang, H. X.; Ning, Q.: The common solution to six quaternion matrix equations with applications, Appl. math. Comput. 198, 209-226 (2008) · Zbl 1141.15016 · doi:10.1016/j.amc.2007.08.091
[18]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 · doi:10.1016/j.laa.2003.12.039
[19]Marsaglia, G.; Styan, G. P. H.: Equalities and inequalities for ranks of matrices, Linear and multilinear algebra 2, 269-292 (1974) · Zbl 0297.15003
[20]Farenick, D. R.; Pidkowich, B. A. F.: The spectral theorem in quaternions, Linear algebra appl. 371, 75-102 (2003) · Zbl 1030.15015 · doi:10.1016/S0024-3795(03)00420-8
[21]Tian, Y.: Equalities and inequalities for traces of quaternionic matrices, Algebras groups geom. 19, No. 2, 181-193 (2002) · Zbl 1167.15308