# Conjugacy and factorization results on matrix groups

Banach Center Publications (1994)

- Volume: 30, Issue: 1, page 203-221
- ISSN: 0137-6934

## Access Full Article

top## Abstract

top## How to cite

topLaffey, Thomas. "Conjugacy and factorization results on matrix groups." Banach Center Publications 30.1 (1994): 203-221. <http://eudml.org/doc/262781>.

@article{Laffey1994,

abstract = {In this survey paper, we present (mainly without proof) a number of results on conjugacy and factorization in general linear groups over fields and commutative rings. We also present the additive analogue in matrix rings of some of these results. The first section deals with the question of expressing elements in the commutator subgroup of the general linear group over a field as (simple) commutators. In Section 2, the same kind of problem is discussed for the general linear group over a commutative ring. In Section 3, the analogous question for additive commutators is discussed. The case of integer matrices is given special emphasis as this is an area of current interest. In Section 4, factorizations of an element A ∈ GL(n,F) (F a field) in which at least one of the factors preserves some form (e.g. is symmetric or skew-symmetric) is considered. An application to the size of abelian subgroups of finite p-groups is presented. In Section 5, a curious interplay between additive and multiplicative commutators in $M_n(F)$ (F a field) is identified for matrices of small size and a general factorization theorem for a polynomial using conjugates of its companion matrix is presented.},

author = {Laffey, Thomas},

journal = {Banach Center Publications},

keywords = {survey paper; conjugacy; factorization; linear groups; commutative rings; matrix rings; commutator subgroup; additive commutators; multiplicative commutators},

language = {eng},

number = {1},

pages = {203-221},

title = {Conjugacy and factorization results on matrix groups},

url = {http://eudml.org/doc/262781},

volume = {30},

year = {1994},

}

TY - JOUR

AU - Laffey, Thomas

TI - Conjugacy and factorization results on matrix groups

JO - Banach Center Publications

PY - 1994

VL - 30

IS - 1

SP - 203

EP - 221

AB - In this survey paper, we present (mainly without proof) a number of results on conjugacy and factorization in general linear groups over fields and commutative rings. We also present the additive analogue in matrix rings of some of these results. The first section deals with the question of expressing elements in the commutator subgroup of the general linear group over a field as (simple) commutators. In Section 2, the same kind of problem is discussed for the general linear group over a commutative ring. In Section 3, the analogous question for additive commutators is discussed. The case of integer matrices is given special emphasis as this is an area of current interest. In Section 4, factorizations of an element A ∈ GL(n,F) (F a field) in which at least one of the factors preserves some form (e.g. is symmetric or skew-symmetric) is considered. An application to the size of abelian subgroups of finite p-groups is presented. In Section 5, a curious interplay between additive and multiplicative commutators in $M_n(F)$ (F a field) is identified for matrices of small size and a general factorization theorem for a polynomial using conjugates of its companion matrix is presented.

LA - eng

KW - survey paper; conjugacy; factorization; linear groups; commutative rings; matrix rings; commutator subgroup; additive commutators; multiplicative commutators

UR - http://eudml.org/doc/262781

ER -

## References

top- [ALB] A. A. Albert and B. Muckenhoupt, On matrices of trace zero, Michigan J. Math. 4 (1957), 1-3. Zbl0077.24304
- [ALP] J. L. Alperin, Large Abelian subgroups of p-groups, Trans. Amer. Math. Soc. 117 (1965), 10-20. Zbl0132.27204
- [BAL] C. S. Ballantine, Products of positive definite matrices, III, J. Algebra 10 (1968), 174-182; IV, Linear Algebra Appl. 3 (1970), 79-114. Zbl0225.15012
- [B-Y] C. S. Ballantine and E. L. Yip, Congruence and conjunctivity of matrices, Linear Algebra Appl. 32 (1980), 159-198. Zbl0454.15006
- [B-M-S] H. Bass, J. Milnor and J.-P. Serre, Solution of the congruence subgroup problem for $S{L}_{n}$ (n ≥ 3) and $S{p}_{2}n$ (n ≥ 2), IHES Publ. Math. 33 (1967), 59-137.
- [B-G-H] J. Buhler, R. Gupta and J. Harris, Isotropic subspaces for skewforms and maximal Abelian subgroups of p-groups, J. Algebra 108 (1987), 269-279. Zbl0612.20009
- [C-T] J. T. Campbell and E. C. Trouy, When are two elements of GL(2,ℤ) similar, Linear Algebra Appl. 157 (1991), 175-184. Zbl0743.20047
- [C-K1] D. Carter and G. Keller, Elementary expressions for unimodular matrices, Comm. Algebra 12 (1984), 379-389. Zbl0572.20030
- [C-K2] D. Carter and G. Keller, Bounded elementary generation of $S{L}_{n}\left(\theta \right)$, Amer. J. Math. 105 (1983), 673-687.
- [C-L-R] M.-D. Choi, C. Laurie and H. Radjavi, On commutators and invariant subspaces, Linear and Multilinear Algebra 9 (1981), 329-340. Zbl0455.15011
- [C-H] D. Choudhury and R. A. Horn, A complex orthogonal-symmetric analog of the polar decomposition, SIAM J. Algebraic Discrete Methods 8 (1987), 218-225. Zbl0623.15006
- [C-W] G. Cooke and P. J. Weinberger, On the construction of division chains in algebraic number rings, with applications to $S{L}_{2}$, Comm. Algebra 3 (1975), 481-524. Zbl0315.12001
- [D-V] R. K. Dennis and L. N. Vaserstein, On a question of M. Newman on the number of commutators, J. Algebra 118 (1988), 150-161. Zbl0649.20048
- [DJO] D. Ž. Djoković, Product of two involutions, Arch. Math. (Basel) 18 (1967), 582-584.
- [ELL] E. W. Ellers, Classical Groups, in: Generators and Relations in Groups and Geometries, NATO Adv. Sci. Inst. Ser. C, Kluwer, Dordrecht, 1991, 1-45.
- [F] P. A. Fillmore, On similarity and the diagonal of a matrix, Amer. Math. Monthly 76 (1969), 167-169. Zbl0175.02403
- [GAI] F. J. Gaines, Kato-Taussky-Wielandt commutator relations, Linear Algebra Appl. 1 (1968), 127-138. Zbl0155.06601
- [G-L-R] I. Gohberg, P. Lancaster and L. Rodman, Invariant Subspaces of Matrices with Applications, Wiley, New York, 1986. Zbl0608.15004
- [GOW1] R. Gow, The equivalence of an invertible matrix to its transpose, Linear and Multilinear Algebra 8 (1980), 329-336. Zbl0432.15004
- [GOW2] R. Gow, Products of two involutions in classical groups of characteristic 2, J. Algebra 71 (1981), 583-591. Zbl0464.20029
- [G-L] R. Gow and T. J. Laffey, Pairs of alternating forms and products of two skew-symmetric matrices, Linear Algebra Appl. 63 (1984), 119-132. Zbl0557.15016
- [G-T] R. Gow and C. Tamburini, Generation of SL(n,ℤ) by a Jordan unipotent matrix and its transpose, to appear. Zbl0769.20021
- [GRA] D. R. Grayson, SK₁ of an interesting principal ideal domain, J. Pure Appl. Algebra 20 (1981), 157-163. Zbl0467.18004
- [G-P-R] L. Grunenfelder, L. Paré and H. Radjavi, On a commutator theorem of R. C. Thompson, Linear and Multilinear Algebra 16 (1984), 129-131. Zbl0548.15010
- [GRU] F. Grunewald, Solution of the conjugacy problem in certain arithmetic groups, in: Word Problems II, S. I. Adian, W. W. Boone and G. Higman (eds.), North-Holland, 1980, 101-139.
- [GUS] W. Gustafson, Modules and matrices, Linear Algebra Appl. 157 (1991), 3-19. Zbl0736.16001
- [G-H-R] W. Gustafson, P. Halmos and H. Radjavi, Products of involutions, ibid. 13 (1976), 157-162. Zbl0325.15009
- [H-OM] A. J. Hahn and O. T. O'Meara, The Classical Groups and K-theory, Grundlehren Math. Wiss. 291, Springer, New York, 1989.
- [HON] K. Honda, On commutators in finite groups, Comment. Math. Univ. St. Paul. 2 (1953), 9-12. Zbl0052.02204
- [HUP] B. Huppert, Endliche Gruppen I, Springer, Berlin, 1967. Zbl0217.07201
- [KAL] W. van der Kallen, SL(ℂ[x]) does not have bounded word length, in: Proc. Algebraic K-Theory Conf., Lecture Notes in Math. 996, Springer, 1982, 356-361. Zbl0935.20501
- [KAP1] I. Kaplansky, Linear Algebra and Geometry, Allyn and Bacon, 1963.
- [KAP2] I. Kaplansky, Algebraic polar decomposition, SIAM J. Matrix Anal. Appl. 11 (1990), 213-217. Zbl0698.15006
- [LAF1] T. J. Laffey, Algebras generated by two idempotents, Linear Algebra Appl. 35 (1985), 45-53.
- [LAF2] T. J. Laffey, Factorizations of matrices involving symmetric matrices and involutions, in: Current Trends in Matrix Theory, North-Holland, 1987, 175-198.
- [LAF3] T. J. Laffey, Matrix factorization with symmetry properties, in: Applications of Matrix Theory, Clarendon Press, Oxford, 1989, 63-70.
- [LAF4] T. J. Laffey, Factorizations of integer matrices as products of idempotents and nilpotents, Linear Algebra Appl. 120 (1989), 81-94. Zbl0677.15005
- [LAF5] Products of matrices, in: Generators and Relations in Groups and Geometries, NATO Adv. Sci. Inst. Ser. C, Kluwer, Dordrecht, 1991, 95-123.
- [L-M1] T. J. Laffey and E. Meehan, An extension of a factorization theorem of Wedderburn to matrix rings, Linear Algebra Appl. 172 (1992), 243-260. Zbl0760.15009
- [L-M2] T. J. Laffey and E. Meehan, Factorization of polynomials using commuting matrices, ibid., to appear.
- [L-M3] T. J. Laffey and E. Meehan, Factorization of polynomials involving unipotent Jordan blocks, Appl. Math. Lett. 5 (1992), 29-33. Zbl0755.15005
- [L-R] T. J. Laffey and R. Reams, Integral similarity and commutators of integral matrices, Linear Algebra Appl., to appear. Zbl0801.15015
- [L-W] T. J. Laffey and T. T. West, Polynomial commutators, Bull. Irish Math. Soc., to appear.
- [L-MAC] C. G. Latimer and C. C. MacDuffee, A correspondence between classes of ideals and classes of matrices, Ann. of Math. 34 (1933), 313-316. Zbl0006.29002
- [LEN] H. W. Lenstra, Grothendieck groups of Abelian group rings, J. Pure Appl. Algebra 20 (1981), 173-193. Zbl0467.16016
- [LIS1] D. Lissner, Matrices over polynomial rings, Trans. Amer. Math. Soc. 98 (1961), 285-305. Zbl0111.01703
- [LIS2] D. Lissner, Outer product rings, ibid. 116 (1965), 526-535. Zbl0134.03804
- [LIU] K.-M. Liu, Decompositions of matrices into three involutions, Linear Algebra Appl. 111 (1989), 1-24.
- [NEW] M. Newman, Unimodular commutators, Proc. Amer. Math. Soc. 101 (1987), 605-609. Zbl0633.15007
- [OCH] J. Ochoa, Un modelo elementel para las clases de ideales de un anillo algebraico, Rev. Real Acad. Cienc. Madrid 63 (1974), 711-806.
- [REH] H. P. Rehm, On Ochoa's special matrices in matrix classes, Linear Algebra Appl. 17 (1977), 181-188. Zbl0362.15008
- [ROW] L. H. Rowen, Polynomial Identities in Ring Theory, Academic Press, New York, 1980.
- [SOU1] A. R. Sourour, A factorization theorem for matrices, Linear and Multilinear Algebra 19 (1986), 141-147. Zbl0591.15008
- [SOU2] A. R. Sourour, Nilpotent factorization of matrices, ibid. 31 (1992), 303-308. Zbl0754.15009
- [TAU1] O. Taussky, On a theorem of Latimer and MacDuffee, Canad. J. Math. 1 (1949), 300-302. Zbl0045.15404
- [TAU2] O. Taussky, Positive definite matrices and their role in the study of the characteristic roots of general matrices, Adv. in Math. 2 (1967), 175-186. Zbl0197.02702
- [T-Z] O. Taussky and H. Zassenhaus, On the similarity transformation between a matrix and its transpose, Pacific J. Math. 9 (1959), 893-896. Zbl0087.01501
- [THO1] R. C. Thompson, Commutators in the special and general linear groups, Trans. Amer. Math. Soc. 101 (1961), 16-33. Zbl0109.26002
- [THO2] R. C. Thompson, Commutators of matrices with prescribed determinants, Canad. J. Math. 20 (1968), 203-221.
- [TOW] J. Towber, Complete reducibility in exterior algebras over free modules, J. Algebra 10 (1968), 299-309. Zbl0186.34103
- [TRO] S. M. Trott, A pair of generators for the unimodular group, Canad. Math. Bull. 5 (1962), 245-252. Zbl0107.02503
- [VAS] L. N. Vaserstein, Noncommutative number theory, algebraic K-theory and algebraic number theory, in: Contemp. Math. 83, Amer. Math. Soc., 1985, 445-449.
- [V-W] L. N. Vaserstein and E. Wheland, Factorization of invertible matrices over rings of stable rank one, preprint, 1990. Zbl0705.15004
- [WAT] W. C. Waterhouse, Pairs of quadratic forms, Invent. Math. 37 (1976), 157-164. Zbl0337.10015
- [WIL] J. Williamson, The equivalence of non-singular pencils of Hermitian matrices in an arbitrary field, Amer. J. Math. 57 (1935), 475-490. Zbl61.0073.01
- [WON1] M. J. Wonenberger, A decomposition of orthogonal transformations, Canad. Math. Bull. 7 (1964), 379-383.
- [WON2] M. J. Wonenberger, Transformations which are products of two involutions, J. Math. Mech. 16 (1966), 327-338.
- [WU1] P. Y. Wu, Products of nilpotent matrices, Linear Algebra Appl. 96 (1987), 227-232. Zbl0628.15008
- [WU2] P. Y. Wu, The operator factorization theorems, ibid. 117 (1989), 35-63.

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.