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.