# Equivalence of Definitions of Matrix Similarity

## Theorem

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 $R$.

The following definitions of the concept of Matrix Similarity are equivalent:

### Definition 1

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.

### Definition 2

$\mathbf A$ and $\mathbf B$ are similar if and only if they are the relative matrices, to (possibly) different ordered bases, of the same linear operator.

## Proof

This is specifically demonstrated in the corollary to Change of Basis Matrix under Linear Transformation.

$\blacksquare$