Given 
, let its QR decomposition be given as 
,
where 
is an 
orthonormal matrix and 
is upper
triangular.
To minimize 
, we choose
m - n more orthonormal vectors (represented by the matrix 
) and
add them as columns to 
such that
is an orthonormal square matrix. In
particular, 
is a matrix obtained by merging
the columns of 
and 
.
Now
Choose 
, such that
, (if 
is
non-singular)
Comparison with Normal Equation Solution
Cost: 
, about 
normal equations if m >> n.