Editing Smith normal form (section)

{{Short description|Matrix normal form}}
In [[mathematics]], the '''Smith normal form''' (sometimes abbreviated '''SNF'''<ref>{{cite journal|last=Stanley|first=Richard P.|authorlink=Richard P. Stanley|date=2016|title=Smith normal form in combinatorics|journal=[[Journal of Combinatorial Theory]] | series=Series A |volume=144|pages=476–495|doi=10.1016/j.jcta.2016.06.013|doi-access=free|arxiv=1602.00166|s2cid=14400632}}</ref>) is a [[Canonical form|normal form]] that can be defined for any [[matrix (mathematics)|matrix]] (not necessarily [[square matrix|square]]) with entries in a [[principal ideal domain]] (PID).  The Smith normal form of a matrix is [[diagonal matrix|diagonal]], and can be obtained from the original matrix by multiplying on the left and right by [[invertible matrix|invertible]] square matrices.  In particular, the [[Integer#Algebraic properties|integers]] are a PID, so one can always calculate the Smith normal form of an [[integer matrix]].  The Smith normal form is very useful for working with [[finitely generated module|finitely generated]] [[module (mathematics)|module]]s over a PID, and in particular for deducing the structure of a [[quotient module|quotient]] of a [[free module]]. It is named after the Irish mathematician [[Henry John Stephen Smith]].<ref>Lazebnik, F. (1996). On systems of linear diophantine equations. Mathematics Magazine, 69(4), 261-266.</ref><ref>Smith, H. J. S. (1861). Xv. on systems of linear indeterminate equations and congruences. Philosophical transactions of the royal society of london, (151), 293-326.</ref>