Equivalence of Definitions of Matrix Equivalence

Theorem

Let $R$ be a ring with unity.

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

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

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.

Proof

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

$\blacksquare$

Sources

• 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {V}$: Vector Spaces: $\S 29$. Matrices