# Categorical description of matrix similarity

I am wondering if there is a nice categorical description of matrix similarity, i.e. the equivalence relation on matrices given by $$A\sim B \iff A=QBQ^{-1}$$, for some invertible $$Q$$.

In particular, I am considering the matrices as linear maps from a vector space into itself, which in turn forms a category $$\mathrm{End}(V)$$ with one object $$V$$ and arrows linear maps. We can then form the functor category $$\mathrm{End}(V)^{\mathbb2},$$ where $$\mathbb2$$ is the category with $$2$$ objects and $$1$$ arrow between them, i.e. this is the 2-category of linear maps and pairs of maps between them making the following commute:

$$\begin{array}{ccccccccc} V & \xrightarrow{A} & V \\ \downarrow & & \downarrow \\ V & \xrightarrow{B} & V \end{array}$$

So similarity between $$A$$ and $$B$$ is stronger than isomorphism in this category, as $$(C,D):A\rightarrow B$$ is an isomorphism if and only if $$C$$ and $$D$$ are linear isomorphisms, whereas similarity would require $$C=D$$. Do similar matrices in this category obey some characterising or otherwise interesting property, or is isomorphism the strongest we can get?

$$\DeclareMathOperator\rank{rank}$$If you allow $$C\neq D$$, an isomorphism $$A\cong B$$ in the category you describe is equivalent to $$\rank(A)=\rank(B)$$. You can of course define a category where as morphisms you only allow commutative squares with $$C=D$$, then an isomorphism $$A\cong B$$ is indeed equivalent to $$A$$ and $$B$$ being similar.
Another way you might want to look at this is in the setting of $$K[X]$$-modules, where $$K$$ is the underlying field. A $$K[X]$$-module amounts to a $$K$$-vector space $$V$$ together with a $$K$$-linear map $$f\colon V\to V$$ that describes the action of $$X$$, that is $$f(v) = X\cdot v$$. In this setting, matrices $$A$$ and $$B$$ yield $$K[X]$$-modules and those are isomorphic if and only if $$A$$ and $$B$$ are similar.