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)
Symmetric and skew-antisymmetric solutions to systems of real quaternion matrix equations. (English) Zbl 1143.15012
Summary: We consider symmetric and skew-antisymmetric solutions to certain matrix equations over the real quaternion algebra $H$. First, a criterion for a quaternion matrix to be symmetric and skew-antisymmetric is given. Then, necessary and sufficient conditions are obtained for the matrix equation $AX=C$ and the following system $$A_1X=C_1, \qquad XB_3=C_3$$ to have symmetric and skew-antisymmetric solutions. The expressions of such solutions of the matrix equation and the system mentioned above are also given.

15A24Matrix equations and identities
15B33Matrices over special rings (quaternions, finite fields, etc.)
Full Text: DOI
[1] Khatri, C. G.; Mitra, S. K.: Hermitian and nonnegative definite solutions of linear matrix equations, SIAM J. Appl. math. 31, 578-585 (1976) · Zbl 0359.65033 · doi:10.1137/0131050
[2] Vetter, W. J.: Vector structures and solutions of linear matrix equations, Linear algebra appl. 9, 181-188 (1975) · Zbl 0307.15003 · doi:10.1016/0024-3795(75)90010-5
[3] Magnus, J. R.; Neudecker, H.: The elimination matrix: some lemmas and applications, SIAM J. Algebr. discrete methods 1, 422-428 (1980) · Zbl 0497.15014 · doi:10.1137/0601049
[4] Don, F. J. Henk: On the symmetric solutions of a linear matrix equation, Linear algebra appl. 93, 1-7 (1987) · Zbl 0622.15001 · doi:10.1016/S0024-3795(87)90308-9
[5] Dai, H.: On the symmetric solution of linear matrix equation, Linear algebra appl. 131, 1-7 (1990) · Zbl 0712.15009 · doi:10.1016/0024-3795(90)90370-R
[6] Navarra, A.; Odell, P. L.; Young, D. M.: A representation of the general common solution to the matrix equations A1XB1=C1 and A2XB2=C2 with applications, Comput. math. Appl. 41, No. 7--8, 929-935 (2001) · Zbl 0983.15016 · doi:10.1016/S0898-1221(00)00330-8
[7] Wang, Q. W.: Bisymmetric and centrosymmetric solutions to systems of real quaternion matrix equations, Comput. math. Appl. 49, 641-650 (2005) · Zbl 1138.15003 · doi:10.1016/j.camwa.2005.01.014
[8] Wang, Q. W.: The general solution to a system of real quaternion matrix equations, Comput. math. Appl. 49, No. 5--6, 665-675 (2005) · Zbl 1138.15004 · doi:10.1016/j.camwa.2004.12.002
[9] 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 · doi:10.1016/S0024-3795(02)00303-8