Equivalent matrices

Definition

Let $F$ be a field and let $A$ and $B$ be $m \times n$ matrices over $F$. The matrix $B$ is equivalent to $A$ when there are an invertible $n \times n$ matrix $P$ and an invertible $m \times m$ matrix $Q$ over $F$ with $$B = Q^{-1} A P.$$

Remarks

Two matrices are equivalent if and only if they represent the same linear map $F^n \to F^m$ in two pairs of bases. Two $m \times n$ matrices over a field are equivalent if and only if they have the same rank. A matrix of rank $r$ is equivalent to the block matrix $\begin{pmatrix} I_r & 0 \\ 0 & 0 \end{pmatrix}$. Similarity is a finer relation: similar matrices are equivalent, but equivalent square matrices need not be similar.

Explanations

L1 human

Matrix equivalence allows independent changes of basis on the input side and on the output side. With so much freedom, only the rank survives. Every matrix of rank $r$ becomes the identity of size $r$, padded with zeros.