A NOTE OF EQUIVALENCE CLASSES OF MATRICES OVER A FINITE FIELD

Let Fqm×m denote the algebra of m×m matrices over the finite field Fq of q elements, and let Ω denote a group of permutations of Fq. It is well known that each ϕϵΩ can be represented uniquely by a polynomial ϕ(x)ϵFq[x] of degree less than q; thus, the group Ω naturally determines a relation ∼ on Fqm×m as follows: if A,BϵFqm×m then A∼B if ϕ(A)=B for some ϕϵΩ. Here ϕ(A) is to be interpreted as substitution into the unique polynomial of degree

F of q elements, and let denote a group of permutations of F It is well q q known that each e can be represented uniquely by a polynomial (x)eF [x] of q degree less than q; thus, the group naturally determines a relation on F mm as follows: if A,BsF mm then AB if (A) B for some e.Here (A) is q q 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 the relation are derived.However, if m > 2, the relation is not an Fmxm.
equivalence relation on It is the purpose of this paper to point out the q 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.

I. INTRODUCTION.
Let F denote the finite field of order q and let F mm denote the algebra of q q mm matrices over Fq.If is a group of permutations of Fq, then can Fmm be used to define a relation on as follows: since each fl can be expressed q uniquely as a polynomial #(x)F [x] of degree <q, we take A to be related to B q (written AB) if B (A) for some where (A) is the substitution of A into the unique polynomial #(x) representing In [I] it is assumed that the relation thus defined is an equivalence rela- tion on F mm 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; namely, the diagonalizable matrices.The q relation, thus restricted, becomes an equivalence relation and again the results of [I] are valid provided suitable enumeration formulas for diagonalizable matrices are known.These needed formulas are derived as a part of section 4.
In [i] it was taken for granted (referring to the relation defined above) -i that if B (A) for some eR then A (B).
To explain why this is not -i necessarily true, consider a particular #fl.Let #(x) and (x) denote the -i -i

q'
This identlt, in F translates to F [x] by the polynomial congruence q q (#-l(x)) -l((x)) x (mod(x q x)).Thus #-l(#(x)) x + h(x)(x q x) for some h(x)eF [x].Hence, if AB by #( le; if (A)= B) we have q #-I(B) -I((A) which is not in general equal to A. Consequently, the relation' is not symmetric.
For a specific example, consider q 5 and take to be the permutation of 3 -I F 5 represented by (x) x Then the polynomial representing # is also (0 (0 0 #-l(x) x Thus for {X,X3}, A 0 0 is related to B 0 as B A 3 however, B is not related to A.

A MODIFICATION OF .
It is quite natural to use the polynomial representation of e to define Fmxm Fmxm the relation on because polynomial functions can act on or any q q algebra over F as easily as on F indeed, this idea is the basis behind q q most notions of extending functions on a field F to functions on F mxm (e.g., see [2]).However, permutation polynomials on F do not in general become permutation polynomials on Fmxm.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 Fmxm obvious linear polynomials ax + b, a # 0, which, when acting on via q substitution do indeed define permutations of Fmxm.These polynomials have q been characterized in [3] where it is shown that f(x) and g(x) represent the same function on F mxmq if and only if f(x) g(x)(rood Lm(X)) where m m i L (x) (x q -x)(x q m x)'"(x q x).
Thus, if G denotes the set of all polynomials of degree less than the degree m m i Fmxm of L (x) ( q + q + + q) which act as permutations on then G m q is a group under composition modulo Lm(X).The group G is studied in [4] where among other things the order IGI is found.
Let be a subgroup of G. Then can be used to define an equivalence re- lation on F mxm by defining AB if there is a e such that B (A).With this q new group playing the role of the group used in [i, section 2], the results and proofs of [I, section 2] are valid.Moreover, if this new is also cyclic, the results and proofs in [I, section 3] are also valid provided the typograph- n/t tluln dbius inversion can of course be applied to the above expression to give M(t,m) where is the classical fdbius function.
Corollary 3.2 of [i] for the number () of equivalence classes of is now valid with the above modifications.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; where (#) is the number of matrix roots of the equation (x) x 0. In [6]   Hodges finds the number of matT,ix roots of f(x) 0, f(x) arbitrary in F [x].
q 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 .simpleexample where this difficulty is readily handled, we take {ax +bla beF a # 0}.Then II q(q i) and it is easily seen (inde- q pendent of Hodges' work) that (ax + b) 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 Clearly (x) is a permutation polynomial on F 5 and ]f[ 4 so that from Theorem 3,1 of [i] we have M(I,2) 624, M(2,2) 0 and M(4,2) i. Thus by Corollary 3.2. of [i] there are_156classes_of order 4 and i class of order I so that %() 157, a result which also follows quite readily from (3.1).

AN ALTERNATE MODIFICATION.
We now cons%der another way to modify the hypotheses in [i] in order to oStain a valid equivalence relation.This time, in contrast to the above alternative, we use the same group of [i] but alter the set on which it is acting, Let be a g=Qup of permutations of F so that each #e is represented by q a unique polynomial #(x) Eai xi e Fq[X] of degree less than q.Let (m,q) de- note the set of m m diagonalizable matrices over Fq; i.e. an m x m matrix A over Fq is in (m,q) if and only if A is similar over F to a diagonal matrix.q 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 q A 0 for all As(m,q).-Hence, the relation defined on.(m,q)by A%B if (A) B for some (x)e is an equivalence relation on m,q).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- ing agreement is made: whenever a formula or statement in [i] calls for the number of matrices AF mm satisfying f(x) 0 where f(x) #(x-x, #(x), we q instead use the number of matrices A(m,q) satisfying f(x) 0. The former number was determined by Hodges in [6] while the latter number is given in the next theorem, the proof of hlch is an adaption of Hodges' methods to our s it uat ion.THEOREM i.Let f(x)eF Ix] have t > 0 distinct roots (with various multi- q pllcties) in F Then the number Z(f(x)) of matrices A(m,q) satisfying q f (A) where the sum is over all t-(kl,...,kt) of nonnegative integers saris- fying Ek i m and where r r -I y(r) (qr l)(q q)-..(q q (4.2) is the well known number 6f invert lhle r x r matrices over F q PROOF.A matrix A is in (m,q) and also satisfies f(x) 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 (all h ,a2 where (al,a2,...,at) is a fixed ordering of the t distinct roots of f(x) and where the klae nonn4gatVe ntegers whose sum is m. (If some k i 0, we understand that the corresponding 51ock does not appear in (4.3)).An m m matrix P is invertible and commutes _th (2.2) if Pt where P i is k i x k i and invertible.Thus from [6] the number of matrices similar to (4.3) is y(m)/(Y(kl)...Y(kt) and the result follows by summing over all (k l,k2,...,kt).
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.
If we take R to be the trivial group fl {#(x) x}, then (4.4) counts the number ID(m,q) of m'x m diagonalizable matrices over F and simplifies using q (4.1) to ID(m,q) r.
y(m) (k l) y (k2)'''y(k q) (4.5) where the sum is overall q-tuples (kl,k2,..., kq) of nonnegative integers with 7.k i m. 3 As a second illustration, if (x) x over F 5 and fl < # > so that RI 2, then as was pointed out in section 2, is a permutation on F 5 but 22 is not an equivalence relation on F 5 However, by restricting R to act on the ID(2,5) 305 d.lagonallzable 22 matrices over F5, we do indeed have a bonafide equivalence relation on D(2,5).Using (4.1) it is not difficult to show that Z(x 3 x) 93 so that by (4.4) l(R) 1/2(93 + 305) 199 distinct equivalence classes.
For R--S (the symmetric group of all permutations of F rather than use q q formula (4.4) which results in a complicated expression, we shall give an independent derivation which utilizes the following theorem whose proof resembles that of Theorem i and will thus be omitted.
Theorem 2. Sl,S2, ,s m Now consider a matrix Ae(m,t) with exactly t distinct eigenvalues, and let P be a fixed matrix such that p-lAp D where D is a diagonal matrix.Let si be the number of eigenvalues of A (or diagonal entries of D) of multiplicity i so that Eis.m and Zs.
t. Hence, associated with A (or D) there is a s12s2 Sm] partition (m,t) [i --.m of m into t parts As (x) runs over S , q (D) runs over all diagonal matrices related to D of which there are clearly -i -I q(q-l)'''(q t + i) q!/t!.Since P(D)P (PDP (A), it follows that the number of matrices related to A is also q!/t! and each such matrix has exactly t distinct eigenvalues.Hence, the number of equivalence classes determined by those matrices with exactly t eigenvalues is t!E(m,t)/q!where m E(m t) is given by (4 6).Thus (S 7. t!E(m,t)/q! is the number of q t=l equivalence classes and this simplifies to to note the similarity in appearance of (4.7) to Cauchy's formula for the number m! of elements in S partitions of m The number of diagonalizable matrices over F with exactly t