For every matrix over , there is a unique matrix in the Howell normal form, such that . The matrix can be obtained from matrix via a sequence of elementary transforms.
From this follows that for two matrices over , their row spans are equal if and only if their Howell normal forms are equal.[2]
For example, the matrices
have the same Howell normal form over :
Note that and are two distinct matrices in the row echelon form, which would mean that their span is the same if they're treated as matrices over some field. Moreover, they're in the Hermite normal form, meaning that their row span is also the same if they're considered over , the ring of integers.[2]
However, is not a field and over general rings it is sometimes possible to nullify a row's pivot by multiplying the row with a scalar without nullifying the whole row. In this particular case,
It implies , which wouldn't be true over any field or over integers.