Showing a set of non-zero matrices that are similar to $A \in M_{n\times n}(\mathbb C)$ are a subspace of $M_{n\times n}(\mathbb C)$

Question

I'm currently studying linear algebra and came across this question that asks to state whether or not if $A \in M_{n\times n}(\mathbb C)$, then the set $S$ containing all $n\times n$ matrices that are similar $A$ is a subspace of $M_{n\times n}(\mathbb C)$

I know that two similar matrices are similar if and only if they have the same Jordan form so the set $S$ contains all $n \times n$ matrices that have the same Jordan form as $A$. But I can't seem to connect the next dot to show whether or not $S$ is a subspace of $M_{n\times n}(\mathbb C)$ or not.

Any hints and guidance will be great

Answer 1

Well, for $A\ne 0$, is the zero matrix $0\in M_{n\times n}(\Bbb{C})$ similar to $A$?

Answer 2

Hint: A slick trick for eliminating certain sets as vector spaces, or subspaces, is to check whether the zero vector $\vec0$ is an element of the set. It must be, because for instance, $0\cdot v=0$ has to be an element of the set. Keep this in mind.

(I still remember when my housemate Kai Behrend, at the time a phd student at Berkeley, pointed this out to me. And I have used it dozens of times. I was already pretty good at linear algebra, but didn't know...)