An overdetermined system is a system of equations containing more equations than unknowns.
In the linear case it has the form 
, where 
is an 
matrix with 
.
An overdetermined system need not be inconsistent. For example, 
and 
have the common solution 
, whereas 
and 
have none. In general, a linear overdetermined system has
an exact solution iff 
belongs to the column
space of 
.
When an exact solution does not exist, the least
squares fitting problem minimizes the squared Euclidean
norm
 |
If the columns of 
are linearly independent,
the solution obtained by least squares fitting
is unique and satisfies the normal equations.
A QR decomposition provides a method for finding
this solution without explicitly forming 
in the real case.
