What are the conjugacy classes in \(\operatorname{GL}_2(\mathbb{C})\)?
Solution
By Jordan canonical form, every element of \(\operatorname{GL}_2(\mathbb C)\) is conjugate to exactly one of the following types, up to permuting diagonal entries:
Distinct eigenvalues: \[ \begin{pmatrix}\lambda&0\\0&\mu\end{pmatrix}, \qquad \lambda,\mu\in\mathbb C^\times, \quad \lambda\ne\mu. \] The class depends only on the unordered pair \(\{\lambda,\mu\}\).
A scalar matrix: \[ \lambda I_2, \qquad \lambda\in\mathbb C^\times. \] Since scalar matrices are central, each such conjugacy class is a singleton.
A nontrivial Jordan block: \[ \begin{pmatrix}\lambda&1\\0&\lambda\end{pmatrix}, \qquad \lambda\in\mathbb C^\times. \]
These are all possibilities because an invertible \(2\times2\) complex matrix has either two distinct nonzero eigenvalues or one repeated nonzero eigenvalue, and in the repeated case it is either diagonalizable or has one Jordan block of size \(2\).