Definition:Matrix Equivalence/Definition 1

Definition

Let $R$ be a ring with unity.

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

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.

We write:

$\mathbf A \equiv \mathbf B$

Also see

• Equivalence of Definitions of Matrix Equivalence

• Results about matrix equivalence can be found here.

Sources

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