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 algorithm for the least squares bisymmetric solutions of the matrix equations $A_{1}XB_{1}=C_{1},A_{2}XB_{2}=C_{2}$. (English) Zbl 1190.65061
The authors propose an iterative algorithm for solving the minimum Frobenius norm residual problem $\min \left[ \left( A_{1}XB_{1}-C_{1} \right)^2 + \left( A_{2}XB_{2}-C_{2} \right)^2 \right]$ over bisymmetric matrices. The algorithm acts on the associated normal equation of the initial one.

65F30Other matrix algorithms
65F10Iterative methods for linear systems
15A24Matrix equations and identities
Full Text: DOI
[1] Mitra, S. K.: Common solutions to a pair of linear matrix equations A1XB1=C1,A2XB2=C2, Proc. Cambridge philos. Soc. 74, 213-216 (1973) · Zbl 0262.15010
[2] Mitra, S. K.: A pair of simultaneous linear matrix equations and a matrix programming problem, Linear algebra appl. 131, 97-123 (1990) · Zbl 0712.15010
[3] Shinozaki, N.; Sibuya, M.: Consistency of a pair of matrix equations with an application, Kieo eng. Rep. 27, 141-146 (1974) · Zbl 0409.15010
[4] J.W. van der Woude, Freeback decoupling and stabilization for linear systems with multiple exogenous variables, Ph.D. Thesis, 1987, pp. 85--199
[5] Navarra, A.; Odell, P. L.; Young, D. M.: A representation of the general common solution to the matrix equations A1XB1=C1,A2XB2=C2 with applications, Comput. math. Appl. 41, 929-935 (2001) · Zbl 0983.15016 · doi:10.1016/S0898-1221(00)00330-8
[6] Yuan, Y. X.: Least squares solutions of matrix equation AXB=E,CXD=F, J. east China shipbuilding inst. 18, No. 3, 29-31 (2004) · Zbl 1098.15009
[7] Deng, Y. B.; Bai, Z. Z.; Gao, Y. H.: Iterative orthogonal direction methods for Hermitian minimum norm solutions of two consistent matrix equations, Numer. linear algebra appl. 13, 801-823 (2006) · Zbl 1174.65382 · doi:10.1002/nla.496
[8] Peng, Y. X.; Hu, X. Y.; Zhang, L.: An iterative method for symmetric solutions and optimal approximation solution of the system of matrix equations, Appl. math. Comput. 183, 1127-1137 (2006) · Zbl 1134.65032 · doi:10.1016/j.amc.2006.05.124
[9] Sheng, X. P.; Chen, G. L.: A finite iterative method for solving a pair of linear matrix equations (AXB,CXD)=(E,F), Appl. math. Comput. 189, 1350-1358 (2007) · Zbl 1133.65026 · doi:10.1016/j.amc.2006.12.026
[10] Meng, T.: Experimental design and decision support, Expert system, the technology of knowledge management and decision making for the 21st century. Vol. 1 1 (2001)
[11] Higham, N. J.: Computing a nearest symmetric positive semidefinite matrix, Linear algebra appl. 103, 103-118 (1988) · Zbl 0649.65026 · doi:10.1016/0024-3795(88)90223-6
[12] Jiang, Z.; Lu, Q.: On optimal approximation of a matrix under a spectral restriction, Math. numer. Sin. 8, 47-52 (1986) · Zbl 0592.65023
[13] Peng, Z. Y.; Hu, X. Y.; Zhang, L.: The inverse problem of bisymmetric matrices, Numer. linear algebra appl. 1, 59-73 (2004) · Zbl 1164.15322 · doi:10.1002/nla.333
[14] Antoniou, A.; Lu, W. S.: Practical optimization: algorithm and engineering applications, (2007) · Zbl 1128.90001
[15] Peng, Z. H.; Hu, X. Y.; Zhang, L.: The bisymmetric solutions of the matrix equation $A1XB1+A2XB2+ \dots +AlXBl=C$ and its optimal approximation, Linear algebra appl. 426, 583-595 (2007) · Zbl 1123.15009 · doi:10.1016/j.laa.2007.05.034
[16] Ben-Israel, A.; Greville, T. N. E.: Generalized inverse: theory and applications, (2002) · Zbl 0451.15004