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By special linear group, I mean matrices with determinant equal to one.

I know a conjugacy class has to be equal to the identity as the identity commutes with other elements. I also know that similar matrices have the same determinant trace and eigenvalues.

How do I find the number of conjugacy classes? and the elements that represent each class? I am familiar with spectral theorem and orbits and stabilizers but don't see how they could be of use.

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  • $\begingroup$ Can you tell us anything about the coefficients of the matrices? Are they complex numbers? Real numbers? Integers? $\endgroup$
    – user729424
    Commented Jan 5, 2020 at 23:30
  • $\begingroup$ Complex matrices $\endgroup$
    – Dhdh
    Commented Jan 5, 2020 at 23:32
  • $\begingroup$ yes @dhdh, they are complex but if someone gave hints on real matrices, I can probably take it from there $\endgroup$
    – John
    Commented Jan 5, 2020 at 23:36
  • $\begingroup$ A conjugacy class is a set of matrices that are conjugate to a particular matrix. You wrote "a conjugacy class has to be equal to the identity". This doesn't make sense because a conjugacy class isn't a matrix; it is a set of matrices. There is one conjugacy class that contains the identity, namely $\{I\}$. The reason there are no other matrices in this set, is because the only matrix that is conjugate to the identity is the identity. $\endgroup$
    – user729424
    Commented Jan 6, 2020 at 0:00

1 Answer 1

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In order to enumerate the conjugacy classes, you must remember that two matrices are conjugate iff they have the same Jordan form. Thus we just have to make an enumeration of all types of Jordan form :

First type : $\begin{equation*} \begin{pmatrix} a & 0 & 0 \\ 0 & b & 0 \\ 0 & 0 & \frac{1}{ab} \end{pmatrix} \end{equation*}$ for each nonzero $(a,b)\in \mathbb{C}^2 $ and up to permutation of the diagonol terms. These are the diagonalisable cases.

Second type : $\begin{equation*} \begin{pmatrix} a & 1 & 0 \\ 0 & a & 0 \\ 0 & 0 & \frac{1}{a} \end{pmatrix} \end{equation*}$ for each nonzero $a \in \mathbb{C} $

Third type : $\begin{equation*} \begin{pmatrix} a & 1 & 0 \\ 0 & a & 1 \\ 0 & 0 & a \end{pmatrix} \end{equation*}$ where $a=1,j$ or $j^2 (j=e^\frac{2i\pi}{3})$

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