Let $\mathbb{K}$ be a field. Two matrices $A,B \in \mathbb{K}^{n \times n} $ are said to be similar if there exists an invertible matrix $S \in \mathbb{K}^{n \times n}$ such that $A = S^{-1}BS$. It can be shown the this relation is an equivalence relation.
My question is: How many equivalence classes of the similarity relation on the set of complex 2×2 matrices, $\mathbb{C}^{n \times n}$, are there? And which ones are they?
My idea would be to use the fact that matrices are similar if and only if they have the same Jordan normal form (up to a permutation of the Jordan blocks). Or is there any another way?
Best Answer
As you said, a similarity class of a matrix $A \in \mathbb C^{n \times n}$ is given by its Jordan normal form, up to a permutation of the Jordan blocks.
As the cardinality of the Jordan normal blocks is the one of the continuum $\mathfrak c$ (i.e. the one of $\mathbb C$), the cardinality you're looking for is $\mathfrak c$.