STATISTICS 512 TECHNIQUES OF MATHEMATICS FOR ...

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q 1 q ), and H v +1 is the “projection of
v +1 onto the space spanned by n q 1 q
o
”.
Thus we say that q +1 is formed by “subtracting
from v +1 its projection onto the space
spanned by n q 1 q
o
”, so as to make what
is left orthogonal to this space (and then normalizing).
• QR-decomposition. In the previous construction,
at each stage, q was obtained as a linear combination
of v 1 v Thus if V × has these vectors
as its columns, and Q × =(q 1 q ), we
can write
V × U × = Q ×
for U upper triangular with positive diagonal
°
elements
( +1+1 = 1 °³
° I
°°
− H´
v+1 0).
Then U is nonsingular and V = QR for R =
U −1 .(NotethatR is also upper triangular with
°
°
positive diagonal elements +1+1 = °³
I
°°.)
− H´
v+1