A unipotent square matrix 
is one for which 
is a nilpotent matrix,
where 
is the identity
matrix. Nilpotence means that 
is the zero matrix for
some positive integer 
. The corresponding identity, 
for some positive
integer 
allows this definition to be generalized to other types of algebraic systems.
An example of a unipotent matrix is a square matrix whose entries below the diagonal are zero and its entries on the diagonal are one. An explicit example of a unipotent matrix is given by
![[1 1 0 0; 0 1 1 0; 0 0 1 1; 0 0 0 1].](/images/equations/Unipotent/NumberedEquation1.svg) |
One feature of a unipotent matrix is that its matrix powers 
have entries which grow like a polynomial in 
.
In a finite-group convention used by Solomon (1995), an element 
of a finite group 
whose element order is a power
of a prime number 
is called unipotent if 
is a p-group,
where 
is the centralizer
of 
and 
is the generalized Fitting
subgroup. This terminology is distinct from calling 
a semisimple element.
