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 iteration method for the symmetric solutions and the optimal approximation solution of the matrix equation $AXB$=$C$. (English) Zbl 1068.65056
Let $\Bbb R^{m\times n}$ be the set of all $m\times n$ matrices, $S\Bbb R^n$ the set of all symmetric matrices in $\Bbb R^{n\times n}$. For $A\in \Bbb R^{m\times n}$, $\Vert A\Vert $ denotes the Frobenius norm. The authors consider the following two problems. Problem 1. Given $A\in \Bbb R^{m\times n}$, $B\in \Bbb R^{n\times p}$, $C\in \Bbb R^{m\times p}$, find $X\in S\Bbb R^{n}$ such that $AXB=C$. Problem 2. If Problem 1 is consistent, then denote its solutions by ${\cal S}_E$. For given $X_0\in \Bbb R^{n\times n}$, find $\hat{X}\in {\cal S}_E$ such that $$ \Vert \hat{X}-X_0\Vert = \min \{\Vert X-X_0\Vert :X\in {\cal S}_E \}. $$ The authors describe an iterative method that determines the solvability of Problem 1 automatically and in the case of solvability computes a solution in an a priori known finite number of steps. Furthermore, the solution to Problem 2 can be found by choosing a suitable initial iteration matrix. It can also be found as the least-norm solution to another equation $A\bar{X}B=\bar{C}$. The paper is carefully written with detailed and convincing proofs. It also contains a numerical example.

65F30Other matrix algorithms
65F10Iterative methods for linear systems
15A24Matrix equations and identities
Full Text: DOI
[1] Golub, G. H.; Van Loan, C. F.: Matrix computations. (1996) · Zbl 0865.65009
[2] Ben-Israel, A.; Greville, T. N. E.: Generalized inverses, theory and applications [M]. (1974) · Zbl 0305.15001
[3] Hua, Dai: On the symmetric solutions of linear matrix equations. Linear algebra appl 131, 1-7 (1990) · Zbl 0712.15009
[4] Chu, King-Wah Eric: Symmetric solutions of linear matrix equations by matrix decompositions. Linear algebra appl 119, 35-50 (1989) · Zbl 0688.15003
[5] Don, F. J. Henk: On the symmetric solution of a linear matrix equation. Linear algebra appl 93, 1-7 (1988)
[6] Magnus, J. R.: L-structured matrices and linear matrix equation. Linear and multilinear algebra appl 14, 67-88 (1983) · Zbl 0527.15006
[7] Morris, G. R.; Odell, P. L.: Common solutions for n matrix equations with applications. J. assoc. Comput. Mach 15, 272-274 (1968) · Zbl 0157.22602
[8] Mitra, S. K.: Common solutions to a pair of linear matrix equations A1XB1=C1, A2XB2=C2. Proc. Cambridge philos., soc 74, 213-216 (1973)
[9] C.R. Rao, Generalized inverse for matrices and its applications in mathematical statistics, Research Papers in Statistics, Festschrift for J. Neyman, John Wiley, New York, 1965
[10] Penrose, R.: A generalized inverse for matrices. Proc. Cambridge philos., soc 51, 406-413 (1955) · Zbl 0065.24603
[11] Bjerhammer, A.: Rectangular reciprocal matrices with special reference to geodetic calculations. Kung tekn. Hogsk. handl. Stockholm 45, 1-86 (1951)