Let $n,s \geq 1$ be integers and $\mathbb{K}$ a field.

We assume there exist $\Phi : \mathcal{M}_n(\mathbb{K}) \rightarrow \mathcal{M}_s(\mathbb{K})$ an **unital** algebra homomorphism ($\Phi(I_n)=I_s$). See here for the definition.

We must show that necessarily $n | s$ and there exists $P \in \textrm{GL}_s(\mathbb{K})$ such that for all $A \in \mathcal{M}_n(\mathbb{K})$ :

$$ P \cdot \Phi(A) \cdot P^{-1} = \begin{pmatrix} A & & (0)\\ & \ddots &\\ (0) & & A \\ \end{pmatrix} $$

I have tried, but I cannot find a way to exploit $\Phi$ to show this. I am looking for a solution of this that uses only "basic" theorems of linear algebra (Bachelor's level). Any help is welcome.