Definition:Matrix Equivalence

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, and we can write $$\mathbf{A} \equiv \mathbf{B}$$.

Thus, from Matrix Corresponding to Change of Basis under Linear Transformation, two matrices are equivalent iff they are the matrices of the same linear transformation, relative to (possibly) different ordered bases.