Two m × n matrices M₁, M₂ are equivalent if there exists an invertible m × m matrix P and an invertible n × n matrix Q such that M₁ = PM₂Q. If the entries are from a field, then a matrix M is equivalent to $\left(\begin{array}{cc}{I}_r& 0\\ 0& 0\end{array}\right)$ where r is the rank of M. See also Smith normal form.