nXn Matrices over GF(q) with a given characteristic polynomial.

[Yuval Dekel]

In a recent thread from the NG sci.math.research named "Nilpotent matrices 
over GF(q)" :
http://mathforum.org/discuss/sci.math/t/554762

Noam Elkeis gave the following :

>Is there a simple proof that the number of nilpotent nXn matrices over
>the finite field GF(q) is q^(n^2-n) ?

The earliest paper in MathSciNet whose review contains
"nilpotent matrices" and "finite field" seems to answer your question:

MR0130875 (24 #A729)
Gerstenhaber, Murray:
On the number of nilpotent matrices with coefficients in a finite field.
Illinois J. Math., Vol.5 (1961), 330--333.

P.Fong's review, stripped of extraneous TeX-ese, reads:

Let GF(q) be the Galois field of q elements, and let GF(q)_n be
the vector space of all n-by-n matrices with coefficients in GF(q).
Fine and Herstein [same J. Vol.2 (1958), 499--504; MR 20 #3160]
have shown that the number of nilpotent matrices in GF(q)_n is q^(n^2-n).
In this article the author gives a simpler proof which runs briefly as 
follows:
Let N be the n-by-n matrix with zeros everywhere except for ones
on the first diagonal above the main diagonal.  For any nilpotent matrix A
in GF(q)_n, let L(A) be the linear subspace of all matrices Y such that 
NY=YA.
The possible matrices Y which can arise are then characterized, and a 
counting
in two ways of the pairs (Y,A), where $A$ is nilpotent and where NY=YA,
together with an induction hypothesis, yields the theorem.
The Fine-Herstein theorem is then used in giving another proof
of a result of Reiner [ibid. Vol.5 (1961), 324--329]
on the number of matrices in GF(q)_n with a given characteristic polynomial.

--Noam D. Elkies


The sequences of nilpotent matrices over GF(q) for q=2,3,4,5
are in the OEIS .

Let a(n)  = number of nXn matrices over GF(q) having characteristic 
polynomial x^n   = 0 and
     b(n)  = number of nXn matrices over GF(q) having characteristic 
polynomial x^n-1 = 0 .

Perhaps someone has access to the paper of Reiner cited above (from 1961) 
and can extract the formula for the number of nXn matrices over GF(q) having 
a given characteristic polynomial to compute
a(n) and b(n) .

Or maybe for small values of q these are in the OEIS ?

Thanks,
Yuval

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