Howell normal form

In linear algebra and ring theory, Howell normal form is a generalization of the row echelon form of a matrix over $\mathbb {Z} _{N}$ ring.

Definition

Let $A\in \mathbb {Z} _{N}^{n\times m}$ be a matrix over $\mathbb {Z} _{N}$ . A matrix is called to be in the row echelon form if it has the following properties:

Let $r$ be the number of non-zero rows of $A$ . Then first $r$ rows of the matrix are non-zero,
For $1\leq i\leq r$ , let $j_{i}$ be the index of a first non-zero element in the row $i$ . Then $i_{1}<i_{2}<\dots <i_{r}$ .

With elementary transforms, any matrix in the row echelon form can be reduced in a way that the following properties will hold:

For any $1\leq i\leq r$ , the leading element of $A_{ij_{i}}$ is a divisor of $m$ ,
For any $1\leq k<i\leq r$ it holds that $0\leq A_{kj_{i}}<A_{ij_{i}}$ .

If matrix adheres to the properties above, it is said to be in the reduced row echelon form.

Let $S(A)$ be the row span of $A$ . Matrix in reduced row echelon form is said to be in the Howell normal form if it adheres to the following additional property:

let $v\in S(A)$ be an elements of the row span of $A$ , such that $v_{k}=0$ for any $1\leq k<i$ . Then $v\in S(A_{i\dots m})$ , where $A_{i\dots m}$ is the matrix obtained of rows from $i$ -th to $m$ -th of the matrix $A$ .

Properties

Let $A,B\in \mathbb {Z} _{N}^{n\times m}$ be matrices over $\mathbb {Z} _{N}$ . Their linear spans are equal if and only if their Howell normal forms are equal. For example, the matrices