I nternat. J. Math. Math. Sci.
Vol. 4 No. 2 (1981) 279-287
279
A NOTE OF EQUIVALENCE CLASSES OF MATRICES
OVER A FINITE FIELD
J.V. BRAWLEY
Department of Mathematical Sciences, Clemson University
Clemson, South Carolina
and
Department of Mathematics, The Pennsylvania State University
Sharon, Pennsylvania
(Received July 16, 1980)
ABSTRACT.
F
q
mxm
Let F
denote the algebra of mm matrices over the finite field
q
denote a group of permutations of F
of q elements, and let
known that each
e
can be represented uniquely by a polynomial
degree less than q; thus, the group
mm
F
q
as follows:
It is well
q
(x)eF q [x] of
naturally determines a relation
mm
then AB if (A)
if A,BsF
q
B for some
e.
on
Here (A) is
to be interpreted as substitution into the unique polynomial of degree <
q which
represents
In an earlier paper by the second author [i], it is assumed that the relation
is an equivalence relation and, based on this assumption, various properties of
However, if m > 2, the relation
the relation are derived.
mxm.
q
equivalence relation on F
is not an
It is the purpose of this paper to point out the
above erroneous assumption, and to discuss two ways in which hypotheses of the
earlier paper can be modified so that the results derived there are valid.
KEY WORDS AND PHRASES.
Equivalence, permutation, automorphism,
980 MATHEMATICS SUBJECT CLASSIFICATION CODES.
I.
inite field.
Prrmary 15A33; Secondary 15A24.
INTRODUCTION.
mm denote the algebra of
denote the finite field of order q and let Fq
q
mm matrices over
If
is a group of permutations of
then
can
Let F
Fq.
Fq,
J.V. BRAWLEY AND G.L. MULLEN
280
be used to define a relation on
Fmm
as
q
follows:
fl can be
since each
expressed
uniquely as a polynomial #(x)F [x] of degree <q, we take A to be related to B
q
where (A) is the substitution of A into
(A) for some
(written AB) if B
the unique polynomial #(x) representing
In [I] it is assumed that the relation thus defined is an equivalence relamm
tion on F
and, based on this assumption, various properties of the relation
q
are derived.
However, for m > 2, the relation is not an equivalence relation.
One of the purposes of this note is to point out the above erroneous assumption.
Another is to discuss ways in which the hypo=heses in [i] can be modified so
that the results derived there are valid.
In section 2 we discuss why
is
not an equivalence relation and in section 3 we briefly indicate how the
results of [i] can be made valid after a simple modification of the group
In section 4 we keep fl as defined in [i], but.we restrict the domain on which it
is acting to be a subset of
From;
q
namely, the diagonalizable matrices.
The
relation, thus restricted, becomes an equivalence relation and again the results
of
[I] are valid provided suitable enumeration formulas for diagonalizable
These needed formulas are derived as a part of section 4.
matrices are known.
2.
THE ERRONEOUS ASSUMPTION.
In [i] it was taken for granted (referring to the relation defined above)
that if B
-i
(A) for some eR then A
(B).
necessarily true, consider a particular #fl.
polynomials representing # and #
This identlt, in F
q
-i
To explain why this is not
Let #(x) and
Then for all aeF
q’ (
-i
-i
(x) denote the
-I
(a))
# ((a))
a.
translates to F [x] by the polynomial congruence
q
q
x (mod(x
x)). Thus
(#-l(x)) -l((x))
#-l(#(x))
for some h(x)eF [x].
q
x
+ h(x)(xq
x)
Hence, if AB by #( le; if (A)= B)
#-I(B)
-I((A))
A + h(A)(Aq
A)
we have
(2.2)
281
EQUIVALENCE CLASSES OF MATRICES OVER A FIELD
which
is not in general equal to A.
Consequently, the relation’
is not
symmetric.
For a specific example, consider q
F
5
represented by (x)
#-l(x)
3
x
3
to be the permutation of
Then the polynomial representing #
{X,X3},
Thus for
x
5 and take
(0
A
is related to B
0 0
-I is also
0
(0
0
as
.
however, B is not related to A.
B
A
3.
A MODIFICATION OF
It is quite natural to use the polynomial representation of
the relation
algebra over F
Fmxm
because
q
on
polynomial functions can act on
as easily as on F
q
e to define
Fmxm
or any
q
indeed, this idea is the basis behind
q
mxm
most notions of extending functions on a field F to functions on F
(e.g.,
However, permutation polynomials on F do not in general become
see [2]).
mxm.
permutation polynomials on F
The failure to account for this fact led to
the error in [i] and the reaizatlon of it leads to the following modification.
It is known [3] that there are polynomials f(x)eF [x] in addition to the
q
+ b,
obvious linear polynomials ax
a
# 0, which, when acting
mxm.
q
substitution do indeed define permutations of F
on
Fmxm
via
q
These polynomials have
been characterized in [3] where it is shown that f(x) and g(x) represent the
mxmq
same function on F
g(x)(rood
if and only if f(x)
(xq
Lm (x)
m
-x)(x q
i
m
Lm(X))
x)’"(x q
where
x).
Thus, if G denotes the set of all polynomials of degree less than the degree
of L (x) (
m
q
m
+ qm
i
+ q)
+
is a group under composition modulo
among other things the order
Let
IGI
be a subgroup of G.
which act as permutations on
Lm(X).
Fmxm
q
then G
The group G is studied in [4] where
is found.
Then
can be used to define an equivalence re-
mxm by
defining AB if there is
q
lation on F
new group playing the role of the group
and proofs of [I, section 2] are valid.
the results and proofs in [I, section 3]
a
e
such that B
(A).
With this
used in [i, section 2], the results
Moreover, if this new
is also cyclic,
are also valid provided the typograph-
J.V. BRAWLEY AND G.L. MULLEN
282
ical error in [i, equation (3.1)] is corrected to read
N(E
M(t,m)
dbius
n/t
M(u,m).
l
tluln
inversion can of course be applied to the above expression to give
M(t,m)
where
,m)
E
tluln
()N(E
n/u
,m)
is the classical fdbius function.
Corollary
3.2 of [i] for the number () of equivalence classes of
now valid with the above modifications.
is
However, a straight forward use of
Burnsides lemma (e,g,, see [5, p. 136]) together with the results of Hodges
[6] perhaps give the simplest expression for the number of equivalence classes;
namely,
>,()
i
()
(3.1)
where (#) is the number of matrix roots of the equation (x)
Hodges finds the number of matT,ix roots of f(x)
x
0.
In [6]
0, f(x) arbitrary in F q [x].
We comment that the main difficulty in using the results of [i] as now
comrected or the above formula (3.1) in conjunction with Hodges’ formulas is
that the polynomials f(x) must be known explicity and in factored form.
As a .simple example where
{ax
+bla
this difficulty is readily handled, we take
a # 0}. Then
q
pendent of Hodges’ work) that
beF
II
q(q
i) and it is easily seen (inde-
283
EQUIVALENCE CLASSES OF MATRICES OVER A FIELD
I if a# 1
(ax + b)
0 if
qm2
# 0
i, b
a
0
i, b
if a
so that
2
l()
qm
+ q(_q
q(q
qm2_1 + q
2)
i)
2
q- i
As an illustration of the theory of [i] in the case where fl is a cyclic
subgroup of G, suppose m
2 and consider
<>
where (x)
2x
+
i over F
5.
2x2
and ]f[
Clearly (x) is a permutation polynomial on F 5
4 so that from
i. Thus by
Theorem 3,1 of [i] we have M(I,2)
0 and M(4,2)
624, M(2,2)
Corollary 3.2. of [i] there are_156classes_of order 4 and i class of order I so
that %()
4.
157, a result which also follows quite readily from (3.1).
AN ALTERNATE MODIFICATION.
We now cons%der another way to modify
oStain a valid equivalence relation.
the hypotheses in [i] in order to
in contrast to the above
This time,
alternative, we use the same group of [i] but alter the set on which it is
acting,
be a g=Qup of permutations of F so that each #e is represented by
q
Let
Eaixi e Fq[X]
a unique polynomial #(x)
note the set of m
A over
Fq
Let (m,q) de-
Fq;
m x m matrix
m diagonalizable matrices over
(m,q) if
is in
of degree less than q.
i.e.
and only if A is similar over F
q
an
to a diagonal matrix.
It is shown in [3, Theorem 5] that each (x)e defimes, via substitution, a
permutation of
(m,q).
This also follows easily fom (2.2) since A
for all As(m,q).- Hence, the relation
some
defined
(x)e is an equivalence relation on m,q).
q
A
on.(m,q) by A%B if (A)
0
B for
J.V. BRAWLEY AND G.L. MULLEN
284
For this particular equivalence
relation on
(m,q), results identical
to
those stated in [i] can be deri=ed (using identical proofs) provided the follow-
whenever a formula or statement in [i] calls for the
ing agreement is made:
mm satisfying f(x)
q
number of matrices AF
instead use the number of matrices
0 where f(x)
x, #(x), we
#(x-
A(m,q) satisfying f(x)
0.
The former
number was determined by Hodges in [6] while the latter number is given in the
Hodges’ methods
next theorem, the proof of hlch is an adaption of
to our
s it ua t ion.
Let f(x)eF Ix] have t > 0 distinct roots (with various multiq
THEOREM i.
pllcties) in F
f (A)
q
Then the number Z(f(x)) of matrices A(m,q) satisfying
0 is
z (f (x))
where the sum is over all
fying Ek
t-
.
Y(kl)Y(k)’’’y(kt)
(kl,...,kt)
(4.1)
of nonnegative integers saris-
m and where
i
(qr
y(r)
l)(q r
q)-..(q
r
q
-I
(4.2)
is the well known number 6f invert lhle r x r matrices over F
PROOF.
A matrix A is in (m,q) and also satisfies f(x)
q
0 if and only if the
minimum polynomial of A factors into distinct linear factors and also divides
f(x).
This is equivalent to saying A is similar to a unique diagonal matrix
of the form
dSag
where
(al,a2,...,at)
where the
(allh ,a2 2
a
t
Ikt )
(4.3)
is a fixed ordering of the t distinct roots of f(x) and
klae nonn4gatVe
ntegers whose sum is m.
(If some k
i
understand that the corresponding 51ock does not appear in (4.3)).
0, we
An m
m
285
EQUIVALENCE CLASSES OF MATRICES OVER A FIELD
matrix P is invertible and commutes
Pt
where P
i
is k
i
x k
i
_th (2.2)
and invertible.
y(m)/(Y(kl)...Y(kt)
(k l,k2,...,kt).
similar to (4.3) is
all
if
Thus from [6] the number of matrices
and the result follows by summing over
We should comment that in case f(x) has no roots in F
the sum (4.1) is
q
the empty sum which by convention is zero.
Formula (4.1) can also be used in conjunction with Burnside’s lemma [5] to
give the following number (R) of equivalence classes:
l(R)
Z(%(x)
where
1
T- Z Z(%(x)
(4.4)
x) is given in (4.1).
{#(x)
If we take R to be the trivial group fl
number
x)
ID(m,q)
x}, then (4.4) counts the
of m’x m diagonalizable matrices over F
q
and simplifies using
(4.1) to
ID(m,q)
r.
i
(k2)’’’y(kq)
(kl,k2,..., kq)
where the sum is overall q-tuples
with 7.k
y(m)
(k l) y
(4.5)
of nonnegative integers
m.
As a second illustration, if (x)
x
3
over F
5
and fl
< # > so that
RI
is a permutation on F but
2, then as was pointed out in section 2,
5
22
is not an equivalence relation on F
However, by restricting R to act on
5
the ID(2,5)
305 d.lagonallzable 22 matrices over F5, we do indeed have a
bonafide equivalence relation on
show that Z(x
3
x)
D(2,5).
Using (4.1) it is not difficult to
93 so that by (4.4) l(R)
1/2(93 + 305)
199 distinct
equivalence classes.
For R--Sq (the symmetric group of all permutations of Fq
rather than use
formula (4.4) which results in a complicated expression, we shall give an
independent derivation which utilizes the following theorem whose proof resembles
J.V. BRAWLEY AND G.L. MULLEN
286
that of Theorem i and will thus be omitted.
The number of diagonalizable matrices over F with exactly t
q
Theorem 2.
eigenvalues is
E(m,t)
E
(m
t)
(m)
(q) (s I ,s 2 t
,s m
(i)i(2)
s
m
i
t
Sl,S 2,
,s m
Is
s
2
s
s
[I I 2 2
where the sum is over all partitions (m,t)
t parts (Eis.
s
s
m
.(m)
m
o__f
(4.6)
m
m into exactly
t)where (r) is given in (4 2) and where
i
(qt)
and
are binomial and multinomial coefficients, respectively.
Now consider a matrix Ae(m,t) with exactly t distinct eigenvalues, and
let P be a fixed matrix such that
si
p-lAp
Let
D where D is a diagonal matrix.
be the number of eigenvalues of A (or diagonal entries of D) of multiplicity
m and Zs.
i so that Eis.
partition (m,t)
[i
s12s2 Sm]
--.m
(D) runs over all diagonal
q(q- l)’’’(q
t
Hence, associated with A (or D) there is
t.
+ i)
a
,
As (x) runs over S
q
of m into t parts
matrices related to D of which there are clearly
q!/t!.
Since
P(D)P
-i
(PDP
-I
(A), it follows
that the number of matrices related to A is also q!/t! and each such matrix has
Hence, the number of equivalence classes
exactly t distinct eigenvalues.
determined by those matrices with exactly t eigenvalues is t!E(m,t)/q! where
m
7. t!E(m,t)/q! is the number of
E(m t) is given by (4 6). Thus (S q
t=l
equivalence classes and this simplifies to
I(S
q
y(m)
E
s
(i)
1
s
Sl!(2) s2!’’’(m)
[i
where the sum is over all partitions
s
2
s12s2 Sm]
---m
(4.7)
msm!
of m
It is interesting to note the similarity in appearance of (4.7) to Cauchy’s
formula for the number m! of elements in S
m
namely,
m!
i
where
[i
s12s2 Sm]
...m
Sl
Sl!2
s2 s
s
!...m
2
msm"
again ranges overll partitions of m
EQUIVALENCE CLASSES OF MATRICES OVER A FIELD
ACKNOWLEDGED:
A portion of this work was completed while the first author
was a visiting professor in the Department of Mathematics,
University of Tennessee, Knoxville, Tennessee.
REFERENCES
i.
MULLEN, G.L.
2.
LANCASTER, P., Theory of Matrices, Academic Press, New York, 1969.
3.
BRAWLEY, J.V., CARLITZ, L., and LEVINE, J., Scalar Polynomial Functions
on the nn trices Over a Finite Field, Linear Alg. and its Appl.
i0 (1975), 199-217.
4.
BRAWLEY, J.V., The Number of Polynomial
5.
LIU, C.L., Introduction to Combinatorial Mathematics, McGraw Hill,
New York, 1968.
6.
HODGES, J.H., Scalar Polynomial Equations for Matrices Over a Finite
Field, Duke Math. J. 25 (1958), 291-296.
Equivalence Classes of Matrices Over a Finite Field,
Internat. J. Math. & Math. Sci. 2 (1979), 487-491.
Functions Which Permute the
Matrices Over a Finite Field, J. Comb. Theory 21 (1976), 147-154.
287