Howell normal form: Difference between revisions

Revision as of 19:25, 10 February 2022

Revision as of 20:56, 10 February 2022

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{{notability|date=February 2022}}

In linear algebra and ring theory, the '''Howell normal form''' is a generalization of the [[row echelon form]] of a matrix over <math>\Z_N</math>, the [[ring of integers modulo n|ring of integers modulo N]]. The linearrow 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 <math>\Z</math>.<ref>{{harvp|Biasse, Fieker, Hofmann|2017|pages=589}}</ref>

== Definition ==

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== Properties ==

LetFor every matrix <math>A,</math> B \inover <math>\Z_N^{n</math>, \timesthere m}is a unique matrix <math>H</math> bein matricesthe overHowell normal form, such that <math>\Z_NS(A)=S(H)</math>. TheirThe linearmatrix spans<math>H</math> arecan equalbe ifobtained andfrom onlymatrix if<math>A</math> theirvia Howella normalsequence formsof areelementary equaltransforms.

From this follows that for two matrices <math>A, B \in \Z_N^{n \times m}</math> over <math>\Z_N</math>, their row spans are equal if and only if their Howell normal forms are equal.<ref name=":0">{{harvp|Storjohann, Mulders|1998|pages=139-140}}</ref>

For example, the matrices

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0 & 0 & 1

\end{bmatrix}.</math>

Note that <math>A</math> and <math>B</math> 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 [[Hermite normal form]], meaning that their row span is also the same if they're considered over <math>\Z</math>, the [[ring of integers]].<ref name=":0" />

However, <math>\Z_{12}</math> 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,

: <math>3 \cdot \begin{bmatrix}4 & 1 & 0\end{bmatrix} \equiv \begin{bmatrix}0 & 3 & 0\end{bmatrix} \pmod{12}.</math>

== References ==

{{Reflist}}

=== Bibliography ===

* {{Cite Q|Q110879587|ref=Howell}}

* {{Cite Q|Q110879586Q110879587|ref=Storjohann, Mulders{{harvid|Howell|1986}}}}

* {{Cite Q|Q110879586|ref={{harvid|Storjohann, Mulders|1998}}}}

* {{Cite Q|Q110883424|ref={{harvid|Biasse, Fieker, Hofmann|2017}}}}

[[Category:Matrix normal forms]]