\begin{definition}[Definition:Matrix Similarity/Definition 1]
Let $R$ be a ring with unity.
Let $n \in \N_{>0}$ be a natural number.
Let $\mathbf A, \mathbf B$ be square matrices of order $n$ over $R$.
Let there exist an invertible square matrix $\mathbf P$ of order $n$ over $R$ such that $\mathbf B = \mathbf P^{-1} \mathbf A \mathbf P$.
Then $\mathbf A$ and $\mathbf B$ are '''similar'''.
We write:
:$\mathbf A \sim \mathbf B$
\end{definition}