zbMATH — the first resource for mathematics

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.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
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)
Extreme ranks of the solution to a consistent system of linear quaternion matrix equations with an application. (English) Zbl 1124.15010
The authors establish the maximal and minimal ranks of the solution to the consistent system of quaternion matrix equations $A_1X=C_1, A_2X=C_2$, $A_3XB_3=C_3$ and $A_4XB_4=C_4$, recently investigated by {\it Q. W. Wang} [Comput. Math. Appl. 49, 665--675 (2005; Zbl 1138.15004)]. As an application, a necessary and sufficient condition for the invariance of the rank of the general solution to the system mentioned above is presented.

15A24Matrix equations and identities
15A03Vector spaces, linear dependence, rank
15B33Matrices over special rings (quaternions, finite fields, etc.)
Full Text: DOI
[1] Tian, Y.: Completing block matrices with maximal and minimal ranks. Linear algebra appl. 321, 327-345 (2000) · Zbl 0984.15013
[2] Tian, Y.: The minimal rank of the matrix expression A-BX-YC. Missouri J. Math. sci. 14, No. 1, 40-48 (2002) · Zbl 1032.15001
[3] Tian, Y.: The minimal rank completion of a $3\times 3$ partial block matrix. Linear multilinear algebra 50, No. 2, 125-131 (2002)
[4] Tian, Y.: Upper and lower bounds for ranks of matrix expressions using generalized inverses. Linear algebra appl. 355, 187-214 (2002) · Zbl 1016.15003
[5] 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
[6] Tian, Y.: Ranks of solutions of the matrix equation AXB=C. Linear multilinear algebra 51, No. 2, 111-125 (2003)
[7] 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
[8] Tian, Y.: More on maximal and minimal ranks of Schur complements with applications. Appl. math. Comput. 152, No. 3, 675-692 (2004) · Zbl 1077.15005
[9] Mitra, S. K.: Fixed rank solutions of linear matrix equations. Sankhya ser. A 35, 387-392 (1972) · Zbl 0261.15008
[10] Mitra, S. K.: The matrix equations AX=C, XB=D. Linear algebra appl. 59, 171-181 (1984) · Zbl 0543.15011
[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] 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)
[13] 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
[14] 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
[15] Wang, Q. W.: The general solution to a system of real quaternion matrix equations. Comput. math. Appl. 49, 665-675 (2005) · Zbl 1138.15004
[16] 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
[17] Wang, Q. W.: Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations. Comput. math. Appl. 49, 641-650 (2005) · Zbl 1138.15003
[18] Tian, Y.: The solvability of two linear matrix equations. Linear multilinear algebra 48, 123-147 (2000) · Zbl 0970.15005
[19] Bhimasankaram, P.: Common solutions to the linear matrix equations AX=B,XC=D, and EXF=G. Sankhya ser. A 38, 404-409 (1976) · Zbl 0411.15008
[20] Lin, C. Y.; Wang, Q. W.: New solvable 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, No. 5, 755-762 (2005) · Zbl 1087.15018
[21] 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
[22] Marsaglia, G.; Styan, G. P. H.: Equalities and inequalities for ranks of matrices. Linear multilinear algebra 2, 269-292 (1974) · Zbl 0297.15003
[23] Wang, Q. W.: A system of four matrix equations over von Neumann regular rings and its applications. Acta math. Sinica, English series 21, No. 2, 323-334 (2005) · Zbl 1083.15021
[24] 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, No. 12, 655-672 (2005) · Zbl 1104.15014
[25] Hungerford, T. W.: Algebra. (1980) · Zbl 0442.00002