Definition:Matrix Similarity/Definition 1

Definition
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:
 * $\mathbf A \sim \mathbf B$

Also see

 * Equivalence of Definitions of Matrix Similarity