Page 221 - A Course in Linear Algebra with Applications

7.1: Scalar Products in Euclidean Space 205

When n = 3, the assertion of 7.1.6 is just the well-known
fact that the sum of the lengths of two sides of a triangle is
never less than the length of the third side, as can be seen
from the triangle rule of addition for the vectors X and Y.

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Complex matrices and orthogonality in C n
It is possible to define a notion of orthogonality in the
complex vector space C™, a fact that will be important in
Chapter Eight. However, a crucial change in the definition
must be made. To see why a change is necessary, consider the

T
complex vector X = ( 7* )• Then X X = - 1 + 1 = 0.
Since it does not seem reasonable to allow a non-zero vector
to have length zero, we must alter the definition of a scalar
product in order to exclude this phenomenon.
First it is necessary to introduce a new operation on com-
plex matrices. Let A be an m x n matrix over the complex
field C. Define the complex conjugate

A

of A to be the m xn matrix whose (i,j) entry is the complex
conjugate of the (i,j) entry of A. Then define the complex
transpose of A to be the transpose of the complex conjugate

A* = (Af.