EXPRESSING UNIPOTENT MATRICES OVER
RINGS AS PRODUCTS OF INVOLUTIONS
Thomas J. La ey
Let R be a ring with 1 and let Mn (R) be the ring of n n matri es
with entries in R. An element A 2 Mn (R) is alled unipotent if
it is of the form In + N where In is the identity n n matrix and
N is either stri tly upper triangular or stri tly lower triangular.
An element J 2 Mn (R) is alled an involution if J 2 = In . In this
note we prove:
Theorem 1 Let R be a ring and n a positive integer. Then every
unipotent element A in Mn (R) is the produ t of ten involutions.
Proof: We an assume A is upper triangular, say
0
1 a12 a13
1 a23
B0
B.
..
.
A=B
.
B.
0 0
0 0
a1 n
a2 n
1
0
an 1 n
1
C
C
C
C
A
1
Write A = HK where
1 a12 a12 a23 0
1
a
0
B0
23
B
0
1
a
a
a45 0
B0
34
34
B
0
0
1
a
0
B0
45
..
H =B
B ..
.
B.
B.
..
B.
.
B.
0
0 0
24
0
0
0
0
1
0
an 1 n
1
1
C
C
C
C
C
C
C
C
C
C
C
A
25
Unipotent matri es
For j odd, row j of H is of the form
(0; 0; : : : ; 0; 1; aj j +1 ; aj +1 j +2 ; : : : ; 0)
while for j even, it has the form
(0; 0; : : : ; 1; aj j +1 ; 0; : : : ; 0):
Note that K = H 1 A has the form
1 0 k13 0 k24
B0 1
B
..
B
.
B
K=B
..
B
.
B
0
B
k1n
k2n
C
C
C
C
C
kn 2 n C
C
C
A
..
.
..
.
0
0 0
1
0
1
Note that H is expressible as produ t H = H1 H2 , where
0
and
1 a12
B0
1
B
0
B0
B
0
B0
0
H1 = B
B0
B.
B.
B.
B.
.
.
0 0
1
B0
B
B0
B
B0
H2 = B
B0
B.
B.
B.
B.
..
0
0
1
0
0
0
0
0
1
0
0
a34 0 1
0
0
1 a56 ..
.
a34 0 1
0
0
1 a45 0 0
0
1
0 1
0
0C
C
0C
0C
C
0C
C;
.. C
C
.C
. . .. C
. .A
0 1
1
0
0C
C
0C
C
0C
C
0C:
.. C
..
.
.C
C
. . .. C
. .A
0 1
26
IMS Bulletin
40, 1998
Using the fa torization
1
1
1
0
=
;
0 1
0
1
0
1
where the two matri es on the right are involutions, we see that H
is the produ t of four involutions.
Note that
K = JL
where J is the full Jordan blo k
1
0
1 1 0 0
B0 1 1
0 0C
B
.. C
.. ..
C
B
.
.
B
.C
B
.. C
.. ..
B
.
.
.C
C
B
C
B
.. ..
B
.
. 0C
C
B
1 1A
0 0 0 1
and L is of the form
0
1
1
1 l13 l1n
B0 1
1 l24 l2n C
B
.. C
.. ..
B
C
.
.
B
. C
B
.. C
.. ..
B
C
.
.
. C:
B
B
C
.. ..
B
.
.
0C
B
C
1
1A
0 0 0
1
If D is the diagonal matrix diag 1; 1; 1; 1; : : : ; ( 1)n 1
then we de ne the matrix L0 by
0
1 1 l13
l14 1
.. C
B
B0 1
1 l24 . C
B
C
.
.
.
B
..
..
.. C
1
B
C:
L0 = D LD = B
C
..
B ..
.
ln 2 n C
B.
C
0
0
1
1 A
0 0
0
1
27
Unipotent matri es
By a result of Dennis and Vaserstein, L0 is similar to J via an
element T of GL(n; R). In fa t, just take T to be a unipotent
upper-triangular matrix and observe that the linear system
T L0 = JT
is solvable indu tively over every ring. (See [2℄, proof of Lemma
13.)
The desired result then follows from our next result.
Proposition The full unipotent Jordan blo k J is the produ t of
three involutions.
Proof:
Let C be the ompanion matrix of (x
0
C=
B
B
B
B
B
B
B
B
B
B
0
0
0
( 1)n 1
1
0
0
1
0 ..
.
..
1)n , so
1
.
..
0
0C
.. C
C
.C
.. C
.C
C
.. C
.C
C
1A
.
n n3
2 n
Put
v1 = (0; 0; : : : ; 0; 1)T ;
v2 = Cv1 v1 = (0; 0; : : : ; 0; 1; n 1)T ;
n 1 T
v3 = Cv2 v2 = (0; 0; : : : ; 0; 1; n 2;
) ;
v4 = Cv3
..
.
v3 = (0; 0; : : : ; 0; 1; n 3;
and let Q = (v1 v2
n 2
..
.
vn = Cvn 1
2
2
;
n 1
3
)T ;
vn 1 = (1; 1; : : : ; 1)T ;
vn ). Then det Q = (
n
1)( 2 ) and CQ = QJ .
28
IMS Bulletin
40, 1998
So J is similar to C over R.
But C is expressible as the produ t C = C1 C2 , where
0
1
B0
B.
B.
B.
B.
.
C1 = B
B.
B.
B ..
B
0
and
0
0
1
0 ..
.
..
.
..
.
n n 1
3
2 n
0
B 0
B
B 0
B .
.
C2 = B
B .
B .
B ..
B
0
( 1)n
1
0
0
0
1
0
0 1
0 1
0
0 C
.. C
C
. C
.. C
. C
C
.. C
. C
C
0 A
1
1
0
0C
C
0C
.. C
..
.
.C
C:
. . .. C
. .C
C
1A
0
Now C1 is an involution and C2 is the produ t of two involutions
(when n is even, this follows from the fa t that every permutation
is the produ t of two involutions and for n odd, we use a slight
modi ation of this argument). See [3℄ for this fa torization of a
ompanion matrix.
Hen e C and therefore J is the produ t of three involutions.
A well-known result of Gustafson, Halmos and Radjavi, [3℄,
states that if F is a eld and A 2 Mn (F ) is su h that det A =
1, then A is the produ t of four involutions in GL(n; F ), but in
general not the produ t of fewer than four. A number of results on
matri es A whi h are the produ t of three involutions is presented
by Liu, [5℄.
Ishibashi [4℄ has obtained a version of the Gustafson-HalmosRadjavi result for integer matri es. He proves that if n > 2 and
A 2 GL(n; Z), then A is the produ t of 3n + 9 involutions in
GL(n; Z).
29
Unipotent matri es
It an be dedu ed from Bass, [1℄, that if n
GL(n; Z), then
S
0
A = U1 L1U2 L2 0 I
n 2
3 and A
2
where U1 , U2 are unipotent upper-triangular and L1 , L2 are unipotent lower-triangular integer matri es and S 2 GL(2; Z) (see
also [2℄, Lemma 9). Sin e
U1 L1U2 L2 = (U1 L1 U1 1 )(U1 U2 )L2
and U1 U2 is unipotent upper-triangular, we dedu e from the theorem that U1 L1 U2 L2 is the produ t of 30 involutions and applying Ishibashi's result in the ase n = 3, we dedu e that A is the
produ t of 48 involutions. Using Bass's result, Dennis and Vaserstein, [2℄, show that for n suÆ iently large (one an he k that
n 82 will do), every A 2 SL(n; Z) an be written as a produ t
U1 L1 U2 L2 U3 L3 where U1 , U2 , U3 are unipotent upper-triangular
and L1 , L2 , L3 unipotent lower-triangular integer matri es. Note
that
U1 L1 U2 L2 U3 L3 = (U1 L1 U1 1 )((U1 U2 )L2 (U1 U2 ) 1 )(U1 U2 U3 )L3
Using this and the fa t that if det A = 1 and
D := diag( 1; 1; 1; : : : ; 1)
then D is an involution and AD 2 SL(n; Z), Theorem 1 yields:
Theorem 2 Every A 2 GL(n; Z) (n 3) an be written as a
produ t of 48 (or fewer) involutions in GL(n; Z). For n 82, this
number
an be redu ed to 41.
Remark (i) It would be interesting to get best possible bounds.
(ii) Ishibashi, [4℄, shows that for elements in GL(2; Z), no su h
bound is possible.
Referen es
[1℄
H. Bass,
5-60.
K -theory and stable algebra, Publ. Math. IHES. 22 (1964),
30
IMS Bulletin
40, 1998
[2℄
R. K. Dennis and L. N. Vaserstein, On a question of M. Newman on
the number of ommutators, J. Algebra 118 (1988), 150-161.
[3℄
W. H. Gustafson, P. R. Halmos and H. Radjavi, Produ ts of involutions,
Colle tion of arti les dedi ated to Olga Taussky Todd. Linear Algebra
Appl. 13 (1976), 157-162.
[4℄
[5℄
H. Ishibashi,
Involutory
expressions
of
elements
SLn (Z ), Linear Algebra Appl. 219 (1995), 165-177.
in
GLn (Z )
and
Kang-Man Liu, De omposition of matri es into three involutions, Linear
Algebra Appl.
111
(1988), 1-24.
Thomas J. La ey
Department of Mathemati s,
University College Dublin,
Bel eld,
Dublin 4,
Ireland.