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.

Sources

• Seth Warner: Modern Algebra (1965): $\S 29$