User Manual

Section 4: Using Matrix Operations 119
)rows1(
),row1(
rows)(
ˆ












pn
p
q
00
0
gU
V
and Û is an upper-triangular matrix. If this factorization results from including n rows r
m
=
(x
m1
, x
m2
, …, x
mp
, y
m
) for m = 1, 2, ... , n in [X y], consider how to advance to n + 1 rows
by appending row r
n+1
to[X y]:



















 11
10
0
n
T
n
r
V
Q
r
yX
.
The zero rows of V are discarded.
Multiply the (p + 2) × (p + 1) matrix
)1(
)1(
)(
ˆ
1
row
row
rowsp
q
n












r
0
gU
A
by a product of elementary orthogonal matrices, each differing from the identity matrix I
p+2
In only two rows and two columns. For k = 1, 2, ... , p + 1 in turn, the k th orthogonal matrix
acts on the k th and last rows to delete the k th element of the last row to alter subsequent
elements in the last row. The k th orthogonal matrix has the form


































 Cs
sc
1
0
1
1
0
1





where c = cos(θ), s = sin(θ), and θ = tan
-1
(a
p+2,k
/ a
kk
). After p + 1 such factors have been
applied to matrix A, it will look like