HP 15c - Manual (Page 96)

96 Section 4: Using Matrix Operations
96
Any n × p matrix X can be factored as X = Q
T
U, where Q is an n × n orthogonal matrix
characterized by Q
T
= Q
−l
and U is an n × p upper-triangular matrix. The essential property
of orthogonal matrices is that they preserve length in the sense that
.
) r ( ) (
2
2
F
T
T T
T
F
r
r r
Qr Q r
Q Qr r Q




Therefore, if r = y – Xb, it has the same length as
Qr = Qy – QXb = Qy – Ub.
The upper-triangular matrix U and the product Qy can be written as
.
rows) (
rows) (
and
rows) (
rows) (
ˆ
p n
p
p n
p
















f
g
Qy
O
U
U
Then
2
2
2
2
2 2
ˆ
r
F
F
F
F
F F
f
f b U g
Ub Qy
Q r

  
 

with equality when 0 b U g
  ˆ
. In other words, the solution to the ordinary least-squares
problem is any solution to g b U
 ˆ
and the minimal sum of squares is
2
F
f . This is the basis
of all numerically sound least-squares programs.
You can solve the unconstrained least-squares problem in two steps:
1. Perform the orthogonal factorization of the augmented n × (p + 1) matrix
 
V Q y X T 
where Q
T
= Q
−1
, and retain only the upper-triangular factor V, which you can then
partition as
rows) 1 (
row) (1
rows) (
ˆ
 











p n
p
q
0 0
0
g U
V
(1 column)
(p columns)