What is the total number of matrices that are similar to the following matrix?

Question

What is the total number of matrices that are similar to \begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix} over $\mathbb{Z}_7$, that is, the finite field on $7$ elements $\{0,1,2,3,4,5,6\}$?

I know that total number of invertible matrices of order $2$ over $\mathbb{Z}_7$ is $2016$. From this how to proceed?

Answer 1

Two matrices in $M_2(\Bbb F_7)$ are similar if and only if they have equal minimal and equal characteristic polynomial, i.e., $A$ is similar to the given matrix if and only if $\operatorname{tr}(A)=1$ and $\det(A)=0$. Now you can count easily the total number of such $A$. Take all matrices $$ \begin{pmatrix} a & b \cr c & 1-a\end{pmatrix} $$ with $a(1-a)=bc$ in $\Bbb F_7$. One has to avoid double counts.

Answer 2

Call your matrix $D$. Its two eigenvalues, $0$ and $1$, are distinct. Therefore a matrix $A$ is similar to $D$ if and only if it shares the same spectrum with $D$. In turn, it is similar to $D$ if and only if both its rank and trace are equal to $1$. This means $A=uv^T$ for some vectors $u$ and $v$ such that $v^Tu=1$. Normalise $u$ such that the first nonzero entry of $u$ is $1$. Then $A$ must take one of the following two forms, where $x,y,z$ are arbitrary scalars in $\mathbb F_7$: $$ \pmatrix{1\\ x}\pmatrix{1-xy&y}\text{ or } \pmatrix{0\\ 1}\pmatrix{z&1}. $$ These representations are unique and so it's a trivial matter to count them.

Answer 3

Here are the matrices satisfying the conditions in Dietrich Burde's answer.

The count is $(11\times 4) + (5\times 2) + (2\times 1) = 56$

Depending whether there can be $2$ or $1$ or $0$ exchange along the two diagonals (i.e. when numbers are not equal).

• Matrices having $0,1$ in main diagonal

$\begin{pmatrix} 0 & 0 \\ 0 & 1 \end{pmatrix}\quad\begin{pmatrix} 1 & 0 \\ 0 & 0 \end{pmatrix}\quad$

$\begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix}\quad\begin{pmatrix} 0 & 0 \\ 1 & 1 \end{pmatrix}\quad\begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix}\quad\begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}\quad$

$\begin{pmatrix} 0 & 2 \\ 0 & 1 \end{pmatrix}\quad\begin{pmatrix} 0 & 0 \\ 2 & 1 \end{pmatrix}\quad\begin{pmatrix} 1 & 2 \\ 0 & 0 \end{pmatrix}\quad\begin{pmatrix} 1 & 0 \\ 2 & 0 \end{pmatrix}\quad$

$\begin{pmatrix} 0 & 3 \\ 0 & 1 \end{pmatrix}\quad\begin{pmatrix} 0 & 0 \\ 3 & 1 \end{pmatrix}\quad\begin{pmatrix} 1 & 3 \\ 0 & 0 \end{pmatrix}\quad\begin{pmatrix} 1 & 0 \\ 3 & 0 \end{pmatrix}\quad$

$\begin{pmatrix} 0 & 4 \\ 0 & 1 \end{pmatrix}\quad\begin{pmatrix} 0 & 0 \\ 4 & 1 \end{pmatrix}\quad\begin{pmatrix} 1 & 4 \\ 0 & 0 \end{pmatrix}\quad\begin{pmatrix} 1 & 0 \\ 4 & 0 \end{pmatrix}\quad$

$\begin{pmatrix} 0 & 5 \\ 0 & 1 \end{pmatrix}\quad\begin{pmatrix} 0 & 0 \\ 5 & 1 \end{pmatrix}\quad\begin{pmatrix} 1 & 5 \\ 0 & 0 \end{pmatrix}\quad\begin{pmatrix} 1 & 0 \\ 5 & 0 \end{pmatrix}\quad$

$\begin{pmatrix} 0 & 6 \\ 0 & 1 \end{pmatrix}\quad\begin{pmatrix} 0 & 0 \\ 6 & 1 \end{pmatrix}\quad\begin{pmatrix} 1 & 6 \\ 0 & 0 \end{pmatrix}\quad\begin{pmatrix} 1 & 0 \\ 6 & 0 \end{pmatrix}\quad$

• Matrices having $2,6$ in main diagonal

$\begin{pmatrix} 2 & 5 \\ 1 & 6 \end{pmatrix}\quad\begin{pmatrix} 2 & 1 \\ 5 & 6 \end{pmatrix}\quad\begin{pmatrix} 6 & 5 \\ 1 & 2 \end{pmatrix}\quad\begin{pmatrix} 6 & 1 \\ 5 & 2 \end{pmatrix}\quad$

$\begin{pmatrix} 2 & 6 \\ 2 & 6 \end{pmatrix}\quad\begin{pmatrix} 2 & 2 \\ 6 & 6 \end{pmatrix}\quad\begin{pmatrix} 6 & 6 \\ 2 & 2 \end{pmatrix}\quad\begin{pmatrix} 6 & 2 \\ 6 & 2 \end{pmatrix}\quad$

$\begin{pmatrix} 2 & 4 \\ 3 & 6 \end{pmatrix}\quad\begin{pmatrix} 2 & 3 \\ 4 & 6 \end{pmatrix}\quad\begin{pmatrix} 6 & 4 \\ 3 & 2 \end{pmatrix}\quad\begin{pmatrix} 6 & 3 \\ 4 & 2 \end{pmatrix}\quad$

• Matrices having $3,5$ in main diagonal

$\begin{pmatrix} 3 & 1 \\ 1 & 5 \end{pmatrix}\quad\begin{pmatrix} 5 & 1 \\ 1 & 3 \end{pmatrix}\quad$

$\begin{pmatrix} 3 & 4 \\ 2 & 5 \end{pmatrix}\quad\begin{pmatrix} 3 & 2 \\ 4 & 5 \end{pmatrix}\quad\begin{pmatrix} 5 & 4 \\ 2 & 3 \end{pmatrix}\quad\begin{pmatrix} 5 & 2 \\ 4 & 3 \end{pmatrix}\quad$

$\begin{pmatrix} 3 & 5 \\ 3 & 5 \end{pmatrix}\quad\begin{pmatrix} 3 & 3 \\ 5 & 5 \end{pmatrix}\quad\begin{pmatrix} 5 & 5 \\ 3 & 3 \end{pmatrix}\quad\begin{pmatrix} 5 & 3 \\ 5 & 3 \end{pmatrix}\quad$

$\begin{pmatrix} 3 & 6 \\ 6 & 5 \end{pmatrix}\quad\begin{pmatrix} 5 & 6 \\ 6 & 3 \end{pmatrix}\quad$

* Matrices having $4,4$ in main diagonal

$\begin{pmatrix} 4 & 2 \\ 1 & 4 \end{pmatrix}\quad\begin{pmatrix} 4 & 1 \\ 2 & 4 \end{pmatrix}\quad$

$\begin{pmatrix} 4 & 3 \\ 3 & 4 \end{pmatrix}\quad$

$\begin{pmatrix} 4 & 4 \\ 4 & 4 \end{pmatrix}\quad$

$\begin{pmatrix} 4 & 6 \\ 5 & 4 \end{pmatrix}\quad\begin{pmatrix} 4 & 5 \\ 6 & 4 \end{pmatrix}\quad$