# Definition:Matrix Equivalence

## Definition

Let $R$ be a ring with unity.

Let $\mathbf A, \mathbf B$ be $m \times n$ matrices over $R$.

### Definition 1

Let there exist:

an invertible square matrix $\mathbf P$ of order $n$ over $R$
an invertible square matrix $\mathbf Q$ of order $m$ over $R$

such that:

$\mathbf B = \mathbf Q^{-1} \mathbf A \mathbf P$

Then $\mathbf A$ and $\mathbf B$ are equivalent.

### Definition 2

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

We write:

$\mathbf A \equiv \mathbf B$

## Also see

• Results about matrix equivalence can be found here.