Page 231 - Advanced Linear Algebra

Real and Complex Inner Product Spaces 215

c
#~ "

~
we have

if " ~
º# Á " » ~ F

if
º"Á "» " £

Therefore, ~ Á when " £ and so " ~ . Hence,

» ~ º# ÁÃÁ# »
º" ÁÃÁ" » ~ º" ÁÃÁ" c Á » ~ º# ÁÃÁ# c

» then

If # ¤ º# ÁÃÁ# c
Á# »

º" ÁÃÁ" » ~ º# ÁÃÁ# c Á" » ~ º# ÁÃÁ# c

Example 9.4 Consider the inner product space s´%µ of real polynomials, with
inner product defined by

º ²%³Á ²%³» ~ ²%³ ²%³ %
c

Applying the Gram–Schmidt process to the sequence 8 ~ ² Á %Á %Á %Á Ã ³
gives
"²%³ ~

% %
"²%³ ~ % c c h ~ %

%
c
% % % %

" ²%³ ~% c c h c c h % ~% c
3
% % %
c c

% % % % %²%c ³ %

"²%³ ~ % c c h c c h % c c 3 h% c ?
>
4
% % % ²% c ³ %

c c c 3

~% c %

(
and so on. The polynomials in this sequence are at least up to multiplicative
)
constants the Legendre polynomials .
The QR Factorization
The Gram–Schmidt process can be used to factor any real or complex matrix
into a product of a matrix with orthogonal columns and an upper triangular
matrix. Suppose that ( ~ ²# “# “Ä“# ³ is an d matrix with columns

# , where  . The Gram–Schmidt process applied to these columns gives
orthogonal vectors 6 ~ ²" “" “Ä“" ³ for which