.
Preliminaries.
Write the squared length of a vector x as .
The least squares estimator 
solves the problem:
Which linear combination of the covariates is ``closest'' to the observed data, Y?
Define the squared length of the residual vector as the
residual sum of squares (RSS), and 
as RSS/(n-p).
Define an 
matrix to be orthogonal if Q satisfies (for any vector)
Orthogonal transformations preserve length.
We can do a transformation on a regression problem:
becomes
Due to the length preservation of Q and the fact that we are doing a least squares problem means that a solution to one is a solution to the other.
Can we choose Q to make the problem simpler and more stable?
Choose Q so that the upper 
block of 
is upper triangular and the lower 
block is all zeroes.
Now
and only the first part depends on 
, giving
which can be solved recursively without inversion.
