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A square matrix  $  A $
over a ring for which the matrix  $  A - I _ {n} $, 
where  $  n $
is the order of  $  A $, 
is nilpotent, i.e.  $  ( A - I _ {n} )  ^ {n} = 0 $. 
A matrix over a field is unipotent if and only if its [[Characteristic polynomial|characteristic polynomial]] is  $  ( x - 1)  ^ {n} $.

A matrix group is called unipotent if every matrix in it is unipotent. Any unipotent subgroup of  $   \mathop{\rm GL} ( n, F  ) $, 
where  $  F $
is a field, is conjugate in  $   \mathop{\rm GL} ( n, F  ) $
to some subgroup of a special triangular group (Kolchin's theorem). This assertion is also true for unipotent groups over a skew-field, if the characteristic of the latter is either 0 or greater than some  $  \gamma ( n) $.