Page 254 - A Course in Linear Algebra with Applications

238 Chapter Seven: Orthogonality in Vector Spaces

and

= - 3 ^ X 2 + \/2X 3 .

Solving back, we obtain the equations

= v^Fi
X x
= 4 ^ / 3 + V6/3
X 2 Y x Y 2
X 3 = 2VSY X + x/6/2 Y 2 + V2/2 Y 3
Therefore A = QR where

7
fl/y/3 -1/V6 l/v ^
Q = l/>/3 2/\/6 0
\lA/3 -1/V6 - l / A

and
(y/3 4/V3 2^3 \
R = 0 \/6/3 \/6/2 .
\ 0 0 V2/2J

Unitary matrices
We point out, without going through the details, that
there is a version of the Gram-Schmidt procedure applicable
to complex inner product spaces. In this the formulas of 7.3.4
are carried over with minor changes, to reflect the properties
of complex inner products.
There is also a QR-factorization theorem. In this an im-
portant change must be made; the matrix Q which is pro-
duced by the Gram-Schmidt process has the property that
its columns are orthogonal with respect to the complex inner
m
product on C . In the case where Q is square this is equiva-
lent to the equation
Q*Q = In