# 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:

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 if and only if they are the matrices of the same linear transformation, relative to (possibly) different ordered bases.