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)
Products of involutory matrices over rings. (English) Zbl 0836.15006
An involution in a group $G$ is an element $a$ of order 2. Let $a \in GL_n A$ be a matrix over a commutative ring $A$ with 1 such that $(\text{det} a)^2 = 1$. If $a$ is cyclic, it can be written as a product of at most three involutions. The paper proves that when $A$ satisfies the first Bass stable range condition, then $a$ can be written as a product of at most five involutions. If in addition either $n \le 3$ or $n = 4$ and $\text{det} a = - 1$, then $a$ can be written as a product of at most four involutions. When $A$ is a Dedekind ring of arithmetic type, the number of involutions needed to express $a$ is uniformly bounded for any $n \ge 3$. When $A = \bbfC[x]$ is the polynomial ring with complex coefficients, the number of involutions is unbounded for any $n\ge 2$.
15B33Matrices over special rings (quaternions, finite fields, etc.)
15A23Factorization of matrices
Full Text: DOI
[1] Ambrosiewicz, E.: Powers of sets of involutions in linear groups. Demonstratio math. 24, 311-314 (1991) · Zbl 0806.20038
[2] Ambrosiewicz, J.: On the square of sets of linear groups. Rend. sem. Mat. univ. Padova 75, 253-256 (1985) · Zbl 0591.20048
[3] . Linear and multilinear algebra 4, 69 (1976)
[4] Ballantine, C. S.: Products of involutory matrices I. Linear and multilinear algebra 5, 53-62 (1977) · Zbl 0364.15017
[5] Berggren, J.: Finite groups in which every element is conjugate to its inverse. Pacific J. Math. 28, 289-293 (1969) · Zbl 0172.03101
[6] Dennis, R. K.; Vaserstein, L. N.: On a question of M. Newman on the number of commutators. J. algebra 118, 150-161 (1988) · Zbl 0649.20048
[7] Djoković, D.: Products of two involutions. Arch. math. 18, 582-584 (1967) · Zbl 0153.35502
[8] Ellers, E.: Products of two involutory matrices over skew fields. Linear algebra appl. 26, 59-63 (1979) · Zbl 0405.15010
[9] Gustafson, W. H.: On products of involutions. Paul halmos celebrating 50 years of mathematics (1991)
[10] Gustafson, W. H.; Halmos, P. R.; Radjavi, H.: Products of involutions. Linear algebra appl. 13, 157-162 (1976) · Zbl 0325.15009
[11] Hoffman, F.; Paige, E. C.: Products of two involutions in the general linear group. Indiana univ. Math. J. 20, 1017-1020 (1971) · Zbl 0237.20041
[12] Knüppel, F.: $GL{\pm}n(R)$ is 5-reflectional. Abh. math. Sem. univ. Hamburg 61, 41-47 (1991) · Zbl 0747.20025
[13] Knüppel, F.; Nielsen, K.: $SL(V)$ is 4-reflectional. Geom. dedicata. 38, 301-308 (1991) · Zbl 0722.20033
[14] Laffey, T. J.: Factorization of matrices involving symmetric matrices and involutions. Current trends in matrix theory, 175-198 (1987)
[15] Liu, K.: Decomposition of matrices into three involutions. Linear algebra appl. 111, 1-24 (1988) · Zbl 0659.15009
[16] Sourour, A.: A factorization theorem for matrices. Linear and multilinear algebra 19, 141-147 (1986) · Zbl 0591.15008
[17] Vaserstein, L. N.; Wheland, E.: Commutators and companion matrices over rings of stable rank 1. Liner algebra appl. 142, 263-277 (1990) · Zbl 0713.15003
[18] Waterhouse, W. C.: Solution to advanced problem 5876. Amer. math. Monthly 81, 1035 (1974)
[19] Wonnenburger, M.: Transformations which are products of two involutions. J. math. Mech. 16, 327-338 (1966) · Zbl 0168.03403
[20] Wu, P. Y.: The operator factorization problems. Linear algebra appl. 117, 35-63 (1989) · Zbl 0673.47018