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)
An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations $A_{1}XB_{1} = C_{1}, A_{2}XB_{2} = C_{2}$. (English) Zbl 1134.65032
The symmetric solutions of the systems of matrix equations $A_1XB_1=C_1$, $A_2XB_2=C_2$ can not be easily obtained by applying matrix decompositions. The authors are proposing an iterative method to solve systems of matrix equations, where when the system of matrix equations is consistent, and its solution can be obtained within finite iterative steps, and its least-norm solution can be obtained by choosing a special kind of initial iterative matrix. Additionally, its optimal approximation solution to a given matrix can be derived by finding the least-norm symmetric solution of a new system of matrix equations $A_1\hat XB_1=\hat C_1$, $A_2\hat XB_2=\hat C_2$. Finally, the author demonstrates the applicability of the proposed method on systems of matrix equations.

65F30Other matrix algorithms
15A24Matrix equations and identities
65F10Iterative methods for linear systems
Full Text: DOI
[1] Mitra, S. K.: The matrix equations AX=C, XB=D. Linear algebra appl. 59, 171-181 (1984) · Zbl 0543.15011
[2] Mitra, S. K.: A pair of simultaneous linear matrix equations A1XB1=C1, and A2XB2=C2 and a matrix programming problem. Linear algebra appl. 131, 107-123 (1990)
[3] 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, 929-935 (2001) · Zbl 0983.15016
[4] Bhimasankaram, P.: Common solutions to the linear matrix equations AX=B, CX=D, and EXF=G. Sankhya ser. A 38, 404-409 (1976) · Zbl 0411.15008
[5] Van Der Woude, J. W.: On the existence of a common solution X to the matrix equations aixbj=Cij, (i,j)$\in \gamma $. Linear algebra appl. 375, 135-145 (2003) · Zbl 1037.15014
[6] Hestenes, M. R.; Stiefel, E.: Methods of conjugate gradients for solving linear systems. J. res. Nat. bur. Stand. 49, 409-436 (1952) · Zbl 0048.09901
[7] Hestenes, M. R.: Conjugate direction methods in optimization. (1980) · Zbl 0439.49001
[8] Reid, J. K.: On the method of conjugate gradients for the solution of large sparse systems of linear equations. Large sparse sets of linear equations, 231-254 (1971)
[9] Young, D. M.; Jea, K. C.: Generalized conjugate gradient acceleration of nonsymmetrizable iterative methods. Linear algebra appl. 34, 159-194 (1980) · Zbl 0463.65025
[10] Axelsson, O.: Conjugate gradient type methods for unsymmetric and inconsistent systems of linear equations. Linear algebra appl. 29, 1-16 (1980) · Zbl 0439.65020
[11] Golub, Gene H.; Van Loan, Charles F.: Matrix computations. (1996) · Zbl 0865.65009
[12] Peng, Ya-Xin; Hu, Xi-Yan; Chang, Lei: An iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation AXB=C. Appl. math. Comput. 160, 763-777 (2005) · Zbl 1068.65056