Howell normal form
In linear algebra and ring theory, the Howell normal form is a generalization of the row echelon form of a matrix over , the ring of integers modulo N. The row spans of two matrices agree if, and only if, their Howell normal forms agree. The Howell normal form generalizes the Hermite normal form, which is defined for matrices over . Definition 
A matrix  over  is called to be in row echelon form if it has the following properties:

 Let  be the number of non-zero rows of . Then the topmost  rows of the matrix are non-zero,
 For , let  be the index of the leftmost non-zero element in the row . Then . With elementary transforms, each matrix in the row echelon form can be reduced in a way that the following properties will hold:

 For each , the leading element  is a divisor of ,
 For each  it holds that . If  adheres to both above properties, it is said to be in reduced row echelon form. If  adheres to the following additional property, it is said to be in Howell normal form ( denotes the row span of ):

 let  be an element of the row span of , such that  for each . Then , where  is the matrix obtained of rows from -th to -th of the matrix . Properties 
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. 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. 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. References

Bibliography 
 
 
 

Matrix normal forms