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Let $p$ be an odd prime and $\mathbb Z_p$ be the prime field of order $p$. Consider the matrix ring $R=M_n(\mathbb Z_p)$. Is there any method to count the solutions of the equation (in the ring $R$)

$$X^2=I.$$ Where $I$ is the identity matrix?

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3 Answers

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Denote $k:=Z_p$. If $p\ne2$, the matrix $X$ is in one-to-one correspondence with a decomposition $k^n=E_+ \oplus E_-$, where $E_\pm$ is the eigenspace associated with the eigenvalue $\pm1$.

Given the dimension $m$ of $E_+$ ($n-m$ for $E_-$), these decompositions are in one-to-one correspondence with the bases of $k^n$, quotiented by the action of $GL_m\times GL_{n-m}$. Therefore, their number is $$\frac{|GL_n(Z_p)|}{|GL_m(Z_p)|\cdot|GL_{n-m}(Z_p)|}=p^{m(n-m)}\frac{(p-1)\cdots(p^n-1)}{(p-1)\cdots(p^m-1)(p-1)\cdots(p^{n-m}-1)}.$$ Summing up over all the possible $m$'s, the number of solutions to $X^2=I$ is $$\sum_{m=0}^np^{m(n-m)}\frac{(p^{m+1}-1)\cdots(p^n-1)}{(p-1)\cdots(p^{n-m}-1)}.$$

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Your formula is precisely the same as my formula, except you wrote $r=n-m$. –  Jason Starr Dec 4 '12 at 0:18
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Because $p$ is different from $2$, the solutions are the same as direct sum decompositions $\mathbb{Z}_p^{\oplus n} = E_{+1} \oplus E_{-1}$ as $\mathbb{Z}_p$-vector spaces. You can index these by the dimensions, say $r$ and $n-r$. For each, the number of solutions is $$ \frac{(p^n-1)(p^n-p) \cdot \dots \cdot (p^n-p^{r-1})}{(p^r-1)(p^r-p) \cdot \dots \cdot (p^r-p^{r-1})} \cdot p^{r(n-r)} $$ So the final answer is the sum over $r$ from $0$ to $n$ of this factor.

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Didn't you count twice the powers of $p$ ? –  Denis Serre Dec 3 '12 at 17:02
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There are $\displaystyle \sum_{i=0}^n \frac{N(n)}{N(i)N(n-i)}$ solutions to your problem, where $N(r)$ is the number of elements in $GL_r(\mathbb{F}_p)$.

Any such $X$ is invertible, and $X\in GL_n(\mathbb{F}_p)$ is a solution if and only if it is conjugate in $GL_n(\mathbb{F}_p)$ to a diagonal matrix with the first $i$ entries equal to $1$ and last $n-i$ entries equal to $-1$. This explains the number of summands. Now $g\in GL_n(\mathbb{F}_p)$ commutes with such diagonal matrix if and only if it preserves both eigenspaces. Hence, there are $\frac{N(n)}{N(i)N(n-i)}$ elements in the conjugacy class.

The formula can be made even more explicit by substituting a formula for $N(r)$.

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Matrices conjugate to a diagonal matrix with $i$ 1's and $n-i$ $-1$'s on the diagonal form a class of matrices with a fixed rational canonical form. For the general problem of enumerating matrices in $R$ with a fixed rational canonical form, see Theorems 1.10.4 and 1.10.7 of Enumerative Combinatorics, vol. 1, 2nd ed. –  Richard Stanley Dec 4 '12 at 1:41
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