Editing Nilpotent matrix (section)

{{Short description|Mathematical concept in algebra}}
In [[linear algebra]], a '''nilpotent matrix''' is a [[square matrix]] ''N'' such that
:<math>N^k = 0\,</math>
for some positive [[integer]] <math>k</math>.  The smallest such <math>k</math> is called the '''index''' of <math>N</math>,<ref>{{harvtxt|Herstein|1975|p=294}}</ref> sometimes the '''degree''' of <math>N</math>.

More generally, a '''nilpotent transformation''' is a [[linear transformation]] <math>L</math> of a [[vector space]] such that <math>L^k = 0</math> for some positive integer <math>k</math> (and thus, <math>L^j = 0</math> for all <math>j \geq k</math>).<ref>{{harvtxt|Beauregard|Fraleigh|1973|p=312}}</ref><ref>{{harvtxt|Herstein|1975|p=268}}</ref><ref>{{harvtxt|Nering|1970|p=274}}</ref> Both of these concepts are special cases of a more general concept of [[nilpotent|nilpotence]] that applies to elements of [[ring (algebra)|rings]].