- Email: [email protected]
A unification of some matrix factorization results
Linear Algebra and its Applications 431 (2009) 1719–1725
Contents lists available at ScienceDirect
Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa
A unification of some matrix factorization results聻 J.D. Botha Department of Mathematical Sciences, University of South Africa, P.O. Box 392, Pretoria 0003, South Africa
A R T I C L E
I N F O
Article history: Received 22 September 2008 Accepted 4 June 2009 Available online 12 July 2009 Submitted by R.A. Brualdi AMS classification: 15A23 15A33
A B S T R A C T
A unification of some factorization results regarding products of positive-definite matrices, commutators, and products of involutions is presented. This topic was first presented by Sourour for fields with, in the latter two classes, sufficiently many elements in terms of the order of the matrix being factored. The current presentation is valid for matrices over any field with at least four elements, and is independent of the order of the matrix being factored. © 2009 Elsevier Inc. All rights reserved.
Keywords: Unification Positive definite Commutator Involution Finite field Nonderogatory Special LU factorization Special LT factorization Spectrum
1. Introduction The purpose of this paper is to present a unified treatment of matrix factorization results by Ballantine [1,2], on products of positive-definite matrices, Shoda [10] and Thompson [13,14], on commutators, and Gustafson et al. [8], on products of involutions.
聻
The author gratefully acknowledges sabbatical leave granted by the University of South Africa, during which this paper was
written. E-mail address: [email protected] 0024-3795/$ - see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2009.06.006
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J.D. Botha / Linear Algebra and its Applications 431 (2009) 1719–1725
This topic was first presented by Sourour [11] who based his unified treatment on a factorization, A = BC, for nonsingular, nonscalar matrices A, in which the eigenvalues β1 , β 2 , . . . , βn of B and γ1 , γ2 , . . . , γn of C are arbitrary subject to the determinantal condition det A = ni=1 βi γi . This treatment is based on the assumption that the underlying field has sufficiently many elements in terms of the order of the matrix being factored. Another paper on this topic [5], based on a factorization theorem for companion matrices, presents a unified treatment for some of these results which is valid over any field with at least four elements, and is independent of the order of the matrix being factored. However, it excludes the results by Ballantine on products of positive-definite matrices as well as some results on commutators, namely, those by Thompson dealing with commutators with prescribed determinants. (On the other hand, it includes the results in [3,4] on products of, singular and nonsingular, diagonable matrices.) The unified treatment presented here is based on an extension, for fields of order at least four, of the factorization theorem by Sourour [11], and involves the factorization of a nonsingular, nonscalar matrix A as a product of two matrices in which both factors are nonderogatory, in addition to having arbitrary eigenvalues, subject to the determinantal condition. This type of factorization was first presented by Johnson and Zhang [9], for matrices over the complex field, and later generalized to fields with at least four elements in [6]. 2. Preliminary results 2.1. Notation and terminology In what follows, F will denote a field with at least four elements, ei the ith column vector of the identity matrix (the order of which will be clear from the context), [v ]i = ci the ith entry of the column ⎡c ⎤ 1
vector v
= ⎣... ⎦ ∈ F n , and Cn (v ) ∈ Mn (F ) the companion matrix with last column equal to v, i.e. cn
Cn (v )
=
⎡
0
⎢ ⎢ ⎢ ⎢ ⎢ ⎢I ⎣ n −1
c1
⎤
⎥ ⎥ c2 ⎥ ⎥. .. ⎥ ⎥ . ⎦ cn
The diagonal matrix of order n with diagonal entries α1 , . . . , αn (in that order) will be denoted by Diag(α1 , . . . , αn ), and the basic Jordan block of order n associated with the eigenvalue 1 will be denoted by Jn , i.e. ⎡ ⎤ 1 ⎡ ⎤ ⎢ ⎥ .. ⎢ ⎥ α1 . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ .. 1 Diag(α1 , . . . , αn ) = ⎣ ⎥. ⎦ , and Jn = ⎢ ⎢ ⎥ ⎢ ⎥ . .. αn ⎣ ⎦ 1
1
A lower (upper) triangular matrix L (U) is called special if all its subdiagonal (superdiagonal) entries li+1,i (ui,i+1 ), 1 ≤ i ≤ n − 1 are nonzero. By a special LU factorization A = LU is therefore meant one in which both L and U are special. A special LT factorization is such that L is lower triangular, T lower or upper triangular, and both are special. GL(n, F ) denotes the multiplicative group of nonsingular n × n matrices over a field F, and SL(n, F ) the subgroup of GL(n, F ) of matrices with determinant 1. A matrix A ∈ GL(n, F ) is unipotent if A − I is nilpotent; it is an involution if A2 = I; and it is a commutator if it can be expressed as A = BCB−1 C −1 with B, C ∈ GL(n, F ). Similarity of A, B ∈ Mn (F ) is denoted by A ∼ B.
J.D. Botha / Linear Algebra and its Applications 431 (2009) 1719–1725
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2.2. LT factorization The unification that follows will be based on the following factorization theorem, which is a modification of a special case of the result treated in [6].
∈ Mn (F ) is a nonsingular, nonscalar matrix over a field F with at least four elements.
Theorem 1. Suppose A Let d = det A.
(a) If rank(A − dI ) > 1 or rank(A − I ) = 1, then A is similar to a special LT factorization, in which all the diagonal entries of L and T are 1, except possibly
the last entry of T , which is d.
= 1 and d = / 1, then A ∼ L U∗
(b) If rank(A − dI )
0 d
, where L and U are special lower and upper
triangular matrices, respectively, with all diagonal entries equal to 1. Proof. (a) If rank(A − dI ) > 1 and rank(A − I ) > 1, then it follows from [6, Theorem 1] that A is similar to a special LT factorization with T upper triangular, by choosing γn = d and all the βi ’s and remaining γi ’s equal to 1. If rank(A − I ) = 1, then I n− 1 0 , where 0 = / t ∈ F. A∼ T d ten−1 Choosing t A
= / 0, 1 ensures that the following factorization of the previous matrix
∼
−1
Jn−1 enT −1
0 1
J n −1
enT −1 (tIn−1
− J n −1 )
0 , d
yields the desired factorization, since ⎡ 1 ⎤ 1+ k T ek ⎡ k=1 (−1) ⎢ ⎥ 1 .. ⎢ ⎥ ⎢ ⎥ ⎢ −1 . ⎢ i ⎥ ⎢ i +k T ⎥ ⎢ 1 Jn−1 = ⎢ ⎢ k=1 (−1) ek ⎥ = ⎢ ⎢ ⎥ ⎣ ... .. ⎢ ⎥ ⎣ ⎦ . (−1)n+1 n n +k T (− 1 ) e k =1 k
0 1 −1
... ...
0 0 1
... 1
... ... ... ... −1
0 0 0
⎤
⎥ ⎥ ⎥ ⎥ . . .⎦ 1
is special, and
[enT −1 (tIn−1 − Jn−1 )]n−1 = t − 1 = / 0. (b) In this case 2 −n d A∼ 1 If n
0 d
⊕ dIn−2 .
= 2, then d2−n = 1, and
A
1
∼ 1
0 d
1
= d
0 1
1
1−d
0 d
yields the desired factorization. If n 3, let 2 −n 0 d A = ⊕ dIn−3 . 1 d Since det A = 1 and rank(A − I ) > 1 (if n = 3, then d2−n = d−1 = / 1), it follows from [6, Theorem 1] that A ∼ LU, where L and U are special and all the diagonal entries of L and U are 1. Hence
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J.D. Botha / Linear Algebra and its Applications 431 (2009) 1719–1725
A
0 d
LU
∼ 0
=
L enT −1
0 1
U −unT −1
0 , d
where unT −1 is the last row of U, yields the desired factorization.
It is known that [11, Corollary, p. 144] an n × n matrix A, over an arbitrary field F, with det A = 1, may be expressed as a product of at most three unipotent matrices. The following shows that, in addition, the factors may be chosen to be nonderogatory, if F has at least four elements. Corollary 2. Suppose A ∈ Mn (F ) such that det A = 1, where F is a field with at least four elements. Then A is a product of three nonderogatory unipotent matrices. If A is nonscalar, then it is a product of two nonderogatory unipotent matrices. Proof. If A is nonscalar, then, since det A = 1, it follows from Theorem 1(a) that A ∼ LT , where all the diagonal entries of L and T are 1 (hence L and T are unipotent), and L and T are special (hence nonderogatory). If A = α I, α n = 1, write A = (α Jn−1 )Jn , and apply the previous argument to the first factor.
3. Products of positive-definite matrices Theorem 3 [1,2]. Let A be a real or complex n × n matrix. Then (a) A is a product of four positive-definite matrices if and only if det A > 0 and A is not a scalar α I where α is not positive. (b) A is a product of five positive-definite matrices if and only if det A > 0. Proof. (a) The case A = α I, α > 0, follows readily. Assume therefore d = det A > 0 and A is nonscalar. By Theorem 1, A ∼ LX, where L is lower triangular and either X = T , a triangular matrix, or it is of U
0
the form ∗ d , where U is upper triangular, and all the diagonal entries of L, U and T are 1, except possibly the last entry of T , which is d. Choose distinct positive numbers a1 , a2 , . . . , an such that an = 1 and a1 , a2 , . . . , an−1 , d are distinct, and let D = Diag (a1 , a2 , . . . , an ). Since A ∼ (D−1 L)(XD), we may write A = BC, where B
∼ D−1 L ∼ D−1 and C ∼ XD ∼ Diag (a1 , a2 , . . . , an−1 , d) = D (say).
Since D−1 and D are positive definite, it follows from an observation by Taussky [12] that both B and C may be expressed as a product of two positive-definite matrices. The converse is as in [11, Theorem 2(a)]. (b) See [11, Theorem 2(b)].
4. Commutators Theorem 4 [10,13,14]. Let A
∈ SL(n, F ), where F is a field with at least four elements.
(a) Then A is a commutator of matrices in GL(n, F ). (b) If A is nonscalar, then A is a commutator of matrices with arbitrarily prescribed nonzero determinants. Proof. (a) If A is nonscalar, then, by Theorem 1(a), A ∼ LT , where L and T are special and all the diagonal entries of L and T are 1 (since det A = 1). Thus, we may express A = BC, where B and C are nonderogatory and all their eigenvalues are 1. It follows that C ∼ B ∼ B−1 , so that C = XB−1 X −1 for some X ∈ GL(n, F ). Thus, A = BXB−1 X −1 .
J.D. Botha / Linear Algebra and its Applications 431 (2009) 1719–1725
In the case where A B
1723
= α I, α n = 1, take
= Diag (α , α 2 , . . . , α n ) and C = Diag (α n , α n−1 , . . . , α),
so that C ∼ B ∼ B−1 , and apply the previous argument. (b) For given β = / 0 and γ = / 0 in F, we want to express A det C = γ . Let Diag (1, 1, . . . , 1, β), if β = / 1; D= Diag (1, 1, . . . , 1, α −1 , α), if β = γ = 1,
= BCB−1 C −1 such that det B = β and
where 0, 1, α −1 , α are distinct in F (which is possible, since |F | 4). According to Theorem 1(a), A ∼ LT , where L and T are special and all the diagonal entries of L and T are 1 (since det A = 1). Thus, A ∼ (DL)(TD−1 ), so we may express A = BE, where E ∼ B−1 (i.e. E = XB−1 X −1 ), det B = β , and B
∼ Jn−1 ⊕ [β] or B ∼ Jn−2 ⊕ [α −1 ] ⊕ [α].
In both cases, there exists a matrix K, with det K A
= γ det X −1 , commuting with B, so that
= BXB−1 X −1 = B(XK )B−1 (XK )−1 = BCB−1 C −1 ,
where C
= XK, and det C = γ . The result follows similarly if β = 1 and γ = / 1.
Remark. Regarding part (b) of the theorem, the case where A is scalar is also treated by Thompson, in [14, Section 2], and is independent of the rest of his paper. 5. Products of involutions Lemma 5. Jn is a product of two involutions. Proof. Since n
n
n
n
(x − 1)n = 0 xn + 1 xn−1 (−1) + 2 xn−2 (−1)2 + · · · + n (−1)n , ⎡
Jn
∼ Cn (v ), where v =
Write Cn (v ) = BKn , where ⎡ n n+ 1 ⎢ n (−1) ⎢ ⎢ B=⎢ ⎢ ⎢ ⎣ v
n (−1)n n
⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ n ⎢ n −1 ⎥ (− 1 ) ⎥. −⎢ ⎢ n−1 ⎥ ⎢ ⎥ . ⎢ ⎥ . ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ n (−1) 1
⎤ 0
q 1
⎥ ⎥ ⎥ ⎥, 1⎥ ⎥ ⎦
⎡ Kn
=⎣ 1
q
⎤ 1 ⎦
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J.D. Botha / Linear Algebra and its Applications 431 (2009) 1719–1725
⎡
n
⎤
n −1 ⎢ n − 1 (−1) ⎥ ⎢ ⎥
and v
=
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ n n −2 ⎥ ⎢ ⎢ n − 2 (−1) ⎥ ⎥. −⎢ ⎢ ⎥ .. ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ n (− 1) 1
Kn is an involution, and ⎡ ⎤ ⎡ n n n +1 n +1 (− 1 ) 0 ⎦ ⎣ n (−1) B2 = ⎣ n K n −1 v v 1 0 = = In , (−1)n+1 v + Kn−1 v In−1 since, for 1 i n − 1,
n
⎤ 0 ⎦ K n −1
n
[(−1)n+1 v + Kn−1 v ]i = − (−1)n+1 n − i (−1)n−i + i (−1)i =− Thus, B is an involution too.
n (−1)2n−i+1 i
n
+ i (−1)i = 0.
Remark. Since Jn ∼ Jn−1 , instead of the above direct proof, this result may also be deduced from [7, Theorem 1], which states that an n × n nonsingular matrix, over an arbitrary field F, is a product of two involutions if and only if A ∼ A−1 . This particular result was first proved in [15, Theorem 1], for fields with characteristic different from 2. Theorem 6 [8]. Let A be an n × n matrix over a field with at least four elements. If det A a product of at most four involutions. Proof. Suppose A is nonscalar. By Theorem 1, A and either X
= ±1, then A is
∼ LX, where L is a special lower triangular matrix
= T , a special triangular matrix, or it is of the form U∗
0 d
, where U is a special upper
triangular matrix, and all the diagonal entries of L, U and T are 1, except possibly the last entry of T , which is det A = ±1. Thus, we may express A = BC, where B ∼ Jn , and C ∼ Jn or C ∼ Jn−1 ⊕ [−1], depending on whether det A = 1 or −1, respectively. Since [−1] is an involution, and since the involutory property is preserved under similarity, it follows from Lemma 5 that both B and C may be expressed as a product of two involutions. If A = α I, α n = ±1, write A = BC, where B = Diag α 2 , α 4 , . . . , α 2n and C = Diag α 2n−1 , α 2n−3 , . . . , α . Note that, for 1 i 2n − 1,
0 1 0 αi 0 = 1 0 0 α 2n−i αi and both factors are involutions.
α 2n−i 0
,
J.D. Botha / Linear Algebra and its Applications 431 (2009) 1719–1725
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If n is even, then C will be, after a permutation of its diagonal elements, a direct sum of matrices of αi 0 , and the same for B, except that one summand will be the form α 2n−i 0 n α 0 (∗) 2n , 0 α which is an involution. Hence both B and C may be expressed as a product of two involutions. The same applies if n is odd, except that now, instead of (∗), B will have a summand [α 2n ], and C will have a summand [α n ].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
C.S. Ballantine, Products of positive definite matrices. III, J. Algebra 10 (1968) 174–182. C.S. Ballantine, Products of positive definite matrices. IV, Linear Algebra Appl. 3 (1970) 79—114. J.D. Botha, Products of diagonalizable matrices, Linear Algebra Appl., 273 (1998) 65–82. J.D. Botha, Products of diagonalizable matrices over a field of characteristic two, Linear Algebra Appl. 286 (1999) 37–44. J.D. Botha, On a unification result by A.R. Sourour concerning commutators and products of involutions, Linear Algebra Appl. 416 (2006) 872–879. J.D. Botha, Spectrally arbitrary, nonderogatory factorization over a general field, submitted for publication. D.Ž. Djokovic, ´ Product of two involutions, Arch. Math. (Basel) 18 (1967) 582–584. W.H. Gustafson, P.R. Halmos, H. Radjavi, Products of involutions, Linear Algebra Appl. 13 (1976) 157–162. C.R. Johnson, Y. Zhang, Spectrally arbitrary factorization: the nonderogatory case, Textos de Matemática, Série B, Universidade Coimbra, Coimbra, 2006, pp. 79-92. K. Shoda, Einige Sätze über Matrizen, Jap. J. Math. 13 (1936) 361–365. A.R. Sourour, A factorization theorem for matrices, Linear and Multilinear Algebra 19 (1986) 141–147. O. Taussky, Positive-definite matrices and their role in the study of the characteristic roots of general matrices, Adv. Math. 2 (1968) 175–186. R.C. Thompson, Commutators in the special and general linear groups, Trans. Amer. Soc. 101 (1961) 16–33. R.C. Thompson, Commutators of matrices with prescribed determinant, Canad. J. Math. 20 (1968) 203–221. M.J. Wonenburger, Transformations which are products of two involutions, J. Math. Mech. 16 (1966) 327–338.
Contents lists available at ScienceDirect
Linear Algebra and its Applications journal homepage: www.elsevier.com/locate/laa
A unification of some matrix factorization results聻 J.D. Botha Department of Mathematical Sciences, University of South Africa, P.O. Box 392, Pretoria 0003, South Africa
A R T I C L E
I N F O
Article history: Received 22 September 2008 Accepted 4 June 2009 Available online 12 July 2009 Submitted by R.A. Brualdi AMS classification: 15A23 15A33
A B S T R A C T
A unification of some factorization results regarding products of positive-definite matrices, commutators, and products of involutions is presented. This topic was first presented by Sourour for fields with, in the latter two classes, sufficiently many elements in terms of the order of the matrix being factored. The current presentation is valid for matrices over any field with at least four elements, and is independent of the order of the matrix being factored. © 2009 Elsevier Inc. All rights reserved.
Keywords: Unification Positive definite Commutator Involution Finite field Nonderogatory Special LU factorization Special LT factorization Spectrum
1. Introduction The purpose of this paper is to present a unified treatment of matrix factorization results by Ballantine [1,2], on products of positive-definite matrices, Shoda [10] and Thompson [13,14], on commutators, and Gustafson et al. [8], on products of involutions.
聻
The author gratefully acknowledges sabbatical leave granted by the University of South Africa, during which this paper was
written. E-mail address: [email protected] 0024-3795/$ - see front matter © 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.laa.2009.06.006
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J.D. Botha / Linear Algebra and its Applications 431 (2009) 1719–1725
This topic was first presented by Sourour [11] who based his unified treatment on a factorization, A = BC, for nonsingular, nonscalar matrices A, in which the eigenvalues β1 , β 2 , . . . , βn of B and γ1 , γ2 , . . . , γn of C are arbitrary subject to the determinantal condition det A = ni=1 βi γi . This treatment is based on the assumption that the underlying field has sufficiently many elements in terms of the order of the matrix being factored. Another paper on this topic [5], based on a factorization theorem for companion matrices, presents a unified treatment for some of these results which is valid over any field with at least four elements, and is independent of the order of the matrix being factored. However, it excludes the results by Ballantine on products of positive-definite matrices as well as some results on commutators, namely, those by Thompson dealing with commutators with prescribed determinants. (On the other hand, it includes the results in [3,4] on products of, singular and nonsingular, diagonable matrices.) The unified treatment presented here is based on an extension, for fields of order at least four, of the factorization theorem by Sourour [11], and involves the factorization of a nonsingular, nonscalar matrix A as a product of two matrices in which both factors are nonderogatory, in addition to having arbitrary eigenvalues, subject to the determinantal condition. This type of factorization was first presented by Johnson and Zhang [9], for matrices over the complex field, and later generalized to fields with at least four elements in [6]. 2. Preliminary results 2.1. Notation and terminology In what follows, F will denote a field with at least four elements, ei the ith column vector of the identity matrix (the order of which will be clear from the context), [v ]i = ci the ith entry of the column ⎡c ⎤ 1
vector v
= ⎣... ⎦ ∈ F n , and Cn (v ) ∈ Mn (F ) the companion matrix with last column equal to v, i.e. cn
Cn (v )
=
⎡
0
⎢ ⎢ ⎢ ⎢ ⎢ ⎢I ⎣ n −1
c1
⎤
⎥ ⎥ c2 ⎥ ⎥. .. ⎥ ⎥ . ⎦ cn
The diagonal matrix of order n with diagonal entries α1 , . . . , αn (in that order) will be denoted by Diag(α1 , . . . , αn ), and the basic Jordan block of order n associated with the eigenvalue 1 will be denoted by Jn , i.e. ⎡ ⎤ 1 ⎡ ⎤ ⎢ ⎥ .. ⎢ ⎥ α1 . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ .. 1 Diag(α1 , . . . , αn ) = ⎣ ⎥. ⎦ , and Jn = ⎢ ⎢ ⎥ ⎢ ⎥ . .. αn ⎣ ⎦ 1
1
A lower (upper) triangular matrix L (U) is called special if all its subdiagonal (superdiagonal) entries li+1,i (ui,i+1 ), 1 ≤ i ≤ n − 1 are nonzero. By a special LU factorization A = LU is therefore meant one in which both L and U are special. A special LT factorization is such that L is lower triangular, T lower or upper triangular, and both are special. GL(n, F ) denotes the multiplicative group of nonsingular n × n matrices over a field F, and SL(n, F ) the subgroup of GL(n, F ) of matrices with determinant 1. A matrix A ∈ GL(n, F ) is unipotent if A − I is nilpotent; it is an involution if A2 = I; and it is a commutator if it can be expressed as A = BCB−1 C −1 with B, C ∈ GL(n, F ). Similarity of A, B ∈ Mn (F ) is denoted by A ∼ B.
J.D. Botha / Linear Algebra and its Applications 431 (2009) 1719–1725
1721
2.2. LT factorization The unification that follows will be based on the following factorization theorem, which is a modification of a special case of the result treated in [6].
∈ Mn (F ) is a nonsingular, nonscalar matrix over a field F with at least four elements.
Theorem 1. Suppose A Let d = det A.
(a) If rank(A − dI ) > 1 or rank(A − I ) = 1, then A is similar to a special LT factorization, in which all the diagonal entries of L and T are 1, except possibly
the last entry of T , which is d.
= 1 and d = / 1, then A ∼ L U∗
(b) If rank(A − dI )
0 d
, where L and U are special lower and upper
triangular matrices, respectively, with all diagonal entries equal to 1. Proof. (a) If rank(A − dI ) > 1 and rank(A − I ) > 1, then it follows from [6, Theorem 1] that A is similar to a special LT factorization with T upper triangular, by choosing γn = d and all the βi ’s and remaining γi ’s equal to 1. If rank(A − I ) = 1, then I n− 1 0 , where 0 = / t ∈ F. A∼ T d ten−1 Choosing t A
= / 0, 1 ensures that the following factorization of the previous matrix
∼
−1
Jn−1 enT −1
0 1
J n −1
enT −1 (tIn−1
− J n −1 )
0 , d
yields the desired factorization, since ⎡ 1 ⎤ 1+ k T ek ⎡ k=1 (−1) ⎢ ⎥ 1 .. ⎢ ⎥ ⎢ ⎥ ⎢ −1 . ⎢ i ⎥ ⎢ i +k T ⎥ ⎢ 1 Jn−1 = ⎢ ⎢ k=1 (−1) ek ⎥ = ⎢ ⎢ ⎥ ⎣ ... .. ⎢ ⎥ ⎣ ⎦ . (−1)n+1 n n +k T (− 1 ) e k =1 k
0 1 −1
... ...
0 0 1
... 1
... ... ... ... −1
0 0 0
⎤
⎥ ⎥ ⎥ ⎥ . . .⎦ 1
is special, and
[enT −1 (tIn−1 − Jn−1 )]n−1 = t − 1 = / 0. (b) In this case 2 −n d A∼ 1 If n
0 d
⊕ dIn−2 .
= 2, then d2−n = 1, and
A
1
∼ 1
0 d
1
= d
0 1
1
1−d
0 d
yields the desired factorization. If n 3, let 2 −n 0 d A = ⊕ dIn−3 . 1 d Since det A = 1 and rank(A − I ) > 1 (if n = 3, then d2−n = d−1 = / 1), it follows from [6, Theorem 1] that A ∼ LU, where L and U are special and all the diagonal entries of L and U are 1. Hence
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J.D. Botha / Linear Algebra and its Applications 431 (2009) 1719–1725
A
0 d
LU
∼ 0
=
L enT −1
0 1
U −unT −1
0 , d
where unT −1 is the last row of U, yields the desired factorization.
It is known that [11, Corollary, p. 144] an n × n matrix A, over an arbitrary field F, with det A = 1, may be expressed as a product of at most three unipotent matrices. The following shows that, in addition, the factors may be chosen to be nonderogatory, if F has at least four elements. Corollary 2. Suppose A ∈ Mn (F ) such that det A = 1, where F is a field with at least four elements. Then A is a product of three nonderogatory unipotent matrices. If A is nonscalar, then it is a product of two nonderogatory unipotent matrices. Proof. If A is nonscalar, then, since det A = 1, it follows from Theorem 1(a) that A ∼ LT , where all the diagonal entries of L and T are 1 (hence L and T are unipotent), and L and T are special (hence nonderogatory). If A = α I, α n = 1, write A = (α Jn−1 )Jn , and apply the previous argument to the first factor.
3. Products of positive-definite matrices Theorem 3 [1,2]. Let A be a real or complex n × n matrix. Then (a) A is a product of four positive-definite matrices if and only if det A > 0 and A is not a scalar α I where α is not positive. (b) A is a product of five positive-definite matrices if and only if det A > 0. Proof. (a) The case A = α I, α > 0, follows readily. Assume therefore d = det A > 0 and A is nonscalar. By Theorem 1, A ∼ LX, where L is lower triangular and either X = T , a triangular matrix, or it is of U
0
the form ∗ d , where U is upper triangular, and all the diagonal entries of L, U and T are 1, except possibly the last entry of T , which is d. Choose distinct positive numbers a1 , a2 , . . . , an such that an = 1 and a1 , a2 , . . . , an−1 , d are distinct, and let D = Diag (a1 , a2 , . . . , an ). Since A ∼ (D−1 L)(XD), we may write A = BC, where B
∼ D−1 L ∼ D−1 and C ∼ XD ∼ Diag (a1 , a2 , . . . , an−1 , d) = D (say).
Since D−1 and D are positive definite, it follows from an observation by Taussky [12] that both B and C may be expressed as a product of two positive-definite matrices. The converse is as in [11, Theorem 2(a)]. (b) See [11, Theorem 2(b)].
4. Commutators Theorem 4 [10,13,14]. Let A
∈ SL(n, F ), where F is a field with at least four elements.
(a) Then A is a commutator of matrices in GL(n, F ). (b) If A is nonscalar, then A is a commutator of matrices with arbitrarily prescribed nonzero determinants. Proof. (a) If A is nonscalar, then, by Theorem 1(a), A ∼ LT , where L and T are special and all the diagonal entries of L and T are 1 (since det A = 1). Thus, we may express A = BC, where B and C are nonderogatory and all their eigenvalues are 1. It follows that C ∼ B ∼ B−1 , so that C = XB−1 X −1 for some X ∈ GL(n, F ). Thus, A = BXB−1 X −1 .
J.D. Botha / Linear Algebra and its Applications 431 (2009) 1719–1725
In the case where A B
1723
= α I, α n = 1, take
= Diag (α , α 2 , . . . , α n ) and C = Diag (α n , α n−1 , . . . , α),
so that C ∼ B ∼ B−1 , and apply the previous argument. (b) For given β = / 0 and γ = / 0 in F, we want to express A det C = γ . Let Diag (1, 1, . . . , 1, β), if β = / 1; D= Diag (1, 1, . . . , 1, α −1 , α), if β = γ = 1,
= BCB−1 C −1 such that det B = β and
where 0, 1, α −1 , α are distinct in F (which is possible, since |F | 4). According to Theorem 1(a), A ∼ LT , where L and T are special and all the diagonal entries of L and T are 1 (since det A = 1). Thus, A ∼ (DL)(TD−1 ), so we may express A = BE, where E ∼ B−1 (i.e. E = XB−1 X −1 ), det B = β , and B
∼ Jn−1 ⊕ [β] or B ∼ Jn−2 ⊕ [α −1 ] ⊕ [α].
In both cases, there exists a matrix K, with det K A
= γ det X −1 , commuting with B, so that
= BXB−1 X −1 = B(XK )B−1 (XK )−1 = BCB−1 C −1 ,
where C
= XK, and det C = γ . The result follows similarly if β = 1 and γ = / 1.
Remark. Regarding part (b) of the theorem, the case where A is scalar is also treated by Thompson, in [14, Section 2], and is independent of the rest of his paper. 5. Products of involutions Lemma 5. Jn is a product of two involutions. Proof. Since n
n
n
n
(x − 1)n = 0 xn + 1 xn−1 (−1) + 2 xn−2 (−1)2 + · · · + n (−1)n , ⎡
Jn
∼ Cn (v ), where v =
Write Cn (v ) = BKn , where ⎡ n n+ 1 ⎢ n (−1) ⎢ ⎢ B=⎢ ⎢ ⎢ ⎣ v
n (−1)n n
⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ n ⎢ n −1 ⎥ (− 1 ) ⎥. −⎢ ⎢ n−1 ⎥ ⎢ ⎥ . ⎢ ⎥ . ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ n (−1) 1
⎤ 0
q 1
⎥ ⎥ ⎥ ⎥, 1⎥ ⎥ ⎦
⎡ Kn
=⎣ 1
q
⎤ 1 ⎦
1724
J.D. Botha / Linear Algebra and its Applications 431 (2009) 1719–1725
⎡
n
⎤
n −1 ⎢ n − 1 (−1) ⎥ ⎢ ⎥
and v
=
⎢ ⎥ ⎢ ⎥ ⎢ ⎥ n n −2 ⎥ ⎢ ⎢ n − 2 (−1) ⎥ ⎥. −⎢ ⎢ ⎥ .. ⎢ ⎥ ⎢ ⎥ . ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ n (− 1) 1
Kn is an involution, and ⎡ ⎤ ⎡ n n n +1 n +1 (− 1 ) 0 ⎦ ⎣ n (−1) B2 = ⎣ n K n −1 v v 1 0 = = In , (−1)n+1 v + Kn−1 v In−1 since, for 1 i n − 1,
n
⎤ 0 ⎦ K n −1
n
[(−1)n+1 v + Kn−1 v ]i = − (−1)n+1 n − i (−1)n−i + i (−1)i =− Thus, B is an involution too.
n (−1)2n−i+1 i
n
+ i (−1)i = 0.
Remark. Since Jn ∼ Jn−1 , instead of the above direct proof, this result may also be deduced from [7, Theorem 1], which states that an n × n nonsingular matrix, over an arbitrary field F, is a product of two involutions if and only if A ∼ A−1 . This particular result was first proved in [15, Theorem 1], for fields with characteristic different from 2. Theorem 6 [8]. Let A be an n × n matrix over a field with at least four elements. If det A a product of at most four involutions. Proof. Suppose A is nonscalar. By Theorem 1, A and either X
= ±1, then A is
∼ LX, where L is a special lower triangular matrix
= T , a special triangular matrix, or it is of the form U∗
0 d
, where U is a special upper
triangular matrix, and all the diagonal entries of L, U and T are 1, except possibly the last entry of T , which is det A = ±1. Thus, we may express A = BC, where B ∼ Jn , and C ∼ Jn or C ∼ Jn−1 ⊕ [−1], depending on whether det A = 1 or −1, respectively. Since [−1] is an involution, and since the involutory property is preserved under similarity, it follows from Lemma 5 that both B and C may be expressed as a product of two involutions. If A = α I, α n = ±1, write A = BC, where B = Diag α 2 , α 4 , . . . , α 2n and C = Diag α 2n−1 , α 2n−3 , . . . , α . Note that, for 1 i 2n − 1,
0 1 0 αi 0 = 1 0 0 α 2n−i αi and both factors are involutions.
α 2n−i 0
,
J.D. Botha / Linear Algebra and its Applications 431 (2009) 1719–1725
1725
If n is even, then C will be, after a permutation of its diagonal elements, a direct sum of matrices of αi 0 , and the same for B, except that one summand will be the form α 2n−i 0 n α 0 (∗) 2n , 0 α which is an involution. Hence both B and C may be expressed as a product of two involutions. The same applies if n is odd, except that now, instead of (∗), B will have a summand [α 2n ], and C will have a summand [α n ].
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]
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