Finding number of matrices whose square is the identity matrix

how can we find the number of matrices with real entries of size $9 \times 9$ (up to similarity) such that $A^{2}=I$?

I first thought about the following:

Notice $A$ satisfies the polynomial $f(t)=t^{2}-1$ hence its minimal polynomial divides $(t-1)(t+1)$.

So its characteristic polynomial is of the form $p(t)=(t-1)^r(t+1)^j$ where $r+j = 9$, right? Then I'm not sure what to do, I tried to consider the rational canonical form but in order to do this we need to know the minimal polynomial right? because in the rational canonical form the last term in the array is exactly the minimal polynomial, how to find it?

Can you please help?

Solution 1:

The eigenvalues are indeed $\pm 1$, so the equation $A^2=I$ is solved exactly by all $A$ of the form $$ A = C D C^{-1} $$ where $C$ is an arbitrary non-singular $9\times 9$ matrix and $D$ is an arbitrary diagonal matrix with $r$ entries $+1$ and $j$ entries $-1$, $r+j=9$. This set of the solution has several disconnected components labeled by the labels $(r,j)$. Each component has the dimension $80-36=44$, I guess, because among the $80$ a priori possible generators of $SL(9)$, the generators of $SO(r,j)$ don't change the matrix.

Solution 2:

As you note, the minimal polynomial divides $(t-1)(t+1)$. Since the minimal polynomial splits and is square free, that means that the matrix is necessarily diagonalizable. Therefore, you want a diagonalizable matrix with eigenvalues $-1$ and/or $1$. Just pick how many times $1$ is an eigenvalue (from $0$ through $9$) to get all similarity types.

Solution 3:

As pointed out in another answer, $A$ has to have Jordan blocks $A_i$ of the form $\pm I +N$ where $N$ is a matrix with a certain number of $1$'s in the first off-diagonal and the rest zeros. The matrix $N^2$ has a certain number of $1$'s in the second off-diagonal and the rest zeros. It follows that the condition $(A_i)^2 = I$, i.e., $I\pm 2 N+ N^2 =I$ implies $N=0$. Therefore $A$ is similar to a diagonal matrix with all diagonal entries $1$ or $-1$. There are 10 different types of such matrices.