Page 222 - A Course in Linear Algebra with Applications

206 Chapter Seven: Orthogonality in Vector Spaces

For example, if

4 —v/-l 3
A =
1 + ^/^T - 4 1 - J=l J '

then

Usually it is more appropriate to use the complex transpose
when dealing with complex matrices. In many ways the com-
plex transpose behaves like the transpose; for example, there
is the following fact.

Theorem 7.1.7
If A and B are complex matrices, then (AB)* = B*A*.
This follows at once from the equations (AB) — (A)(B)
T T T
and (AB) = B A .
Now let us use the complex transpose to define the com-
n
plex scalar product of vectors X and Y in C ; this is to be

X*Y = xtyi + --- + x ny n,

which is a complex number. Why is this definition any better
than the previous one? The reason is that, if we define the
length of the vector X in the natural way as

2
\\X\\ = VX*X = /|x 1 | 2 + --- + |x n | ,
x
then ||X|| is always a non-negative real number, and it can-
not equal 0 unless X is the zero vector. It is an important
consequence of the definition that Y*X equals the complex
conjugate of X*Y, so the complex scalar product is not sym-
metric in X and Y.