Definition:Matrix Equivalence

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, 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.