Definition:Matrix Equivalence

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

Also see