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 formula_1, 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 formula_2. Definition. A matrix formula_3 over formula_1 is called to be in "row echelon form" if it has the following properties: With elementary transforms, each matrix in the row echelon form can be reduced in a way  that the following properties will hold: If formula_6 adheres to both above properties, it is said to be in "reduced row echelon form". If formula_6 adheres to the following additional property, it is said to be in Howell normal form (formula_19 denotes the row span of formula_6): Properties. For every matrix formula_6 over formula_1, there is a unique matrix formula_32 in the Howell normal form, such that formula_33. The matrix formula_32 can be obtained from matrix formula_6 via a sequence of elementary transforms. From this follows that for two matrices formula_36 over formula_1, 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 formula_39: Note that formula_6 and formula_42 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 formula_2, the ring of integers. However, formula_39 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 formula_46, which wouldn't be true over any field or over integers.