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

Also see

 * Equivalent Matrices have Equal Rank
 * Definition:Matrix Similarity
 * Definition:Matrix Congruence


 * Change of Basis Matrix under Linear Transformation