Page 232 - Advanced Linear Algebra

216 Advanced Linear Algebra

º" ÁÃÁ" » ~ º# ÁÃÁ# »

for all  . In particular,
c
#~ " b " Á

~
where
" ~ if
~ Á H º# Á" » if "£
º" Á" »
In matrix terms,

v Á Ä Á y
x Ä Á {
²# “# “Ä“# ³~²" “" “Ä“" ³x {

Æ
w z
that is, (~ 6) where has orthogonal columns and is upper triangular.
)
6
We may normalize the nonzero columns " of 6 and move the positive
)

constants to . In particular, if ~ " )) for "£ and ~ for "~ , then

v Á Ä Á y
" " " Ä x Á {
²# “# “Ä“# ³~ 6 c c Ä c 7 x {

Æ
w z

and so
(~ 89
where the columns of are orthogonal and each column is either a unit vector
8
or the zero vector and is upper triangular with positive entries on the main
9
are linearly independent, then the
diagonal. Moreover, if the vectors #Á Ã Á #
(
8
columns of are nonzero. Also, if ~ and is nonsingular, then is
8
unitary/orthogonal.
If the columns of ( are not linearly independent, we can make one final
is zero, then we may
adjustment to this matrix factorization. If a column "°
replace this column by any vector as long as we replace the ² Á ³ th entry in 9

by . Therefore, we can take nonzero columns of , extend to an orthonormal

8
basis for the span of the columns of and replace the zero columns of by the
8
8
additional members of this orthonormal basis. In this way, is replaced by a
8
unitary/orthogonal matrix 8 Z and is replaced by an upper triangular matrix 9 Z
9
that has nonnegative entries on the main diagonal.