Let 
be an 
matrix of matrix rank 
over a field.
A rank factorization, also called a full-rank factorization or rank decomposition,
is a factorization
 |
where 
is an 
matrix of full column rank and 
is an 
matrix of full row rank.
A rank factorization can be constructed from the reduced row-echelon form of 
.
Let 
consist of its nonzero rows, and let 
consist of the columns of the original matrix 
whose indices are the columns containing the pivot
elements of 
.
The rows of 
form a basis for the row space,
the columns of 
form a basis for the column space, and 
(Piziak and Odell 1999).
A rank factorization is generally not unique. For any invertible matrix 
of size 
,
 |
is another rank factorization.
