Basics of Hermitian Geometry

In this chapter we generalize the basic results of Euclidean geometry presented in Chapter 6 to vector spaces over the complex numbers. Such a generalization is inevitable, and not simply a luxury. For example, linear maps may not have real eigenvalues, but they always have complex eigenvalues. Furthermore, some very important classes of linear maps can be diagonalized if they are extended to the complexification of a real vector space. This is the case for orthogonal matrices, and, more generally, normal matrices. Also, complex vector spaces are often the natural framework in physics or engineering, and they are more convenient for dealing with Fourier series. However, some complications arise due to complex conjugation. Recall that for any complex number

z ∈

$$\mathbb{C} $$

, if

z

=

x

+i

y

where

x, y ∈

ℝ, we let ℜ

z

=

x

, the real part of

z

, and ℑ

z

=

y

, the imaginary part of

z

. We also denote the conjugate of

z

=

x

+i

y

by

$$\bar{z}$$

=

x-

i

y

, and the absolute value (or length, or modulus) of

z

by

|z|

. Recall that

|z|

2 = z

$$\bar{z}$$

=

x

2

+

y

2

. There are many natural situations where a map ϕ :

E × E

$$\mathbb{C} $$

is linear in its first argument and only semilinear in its second argument, which means that ϕ(

u,

µ

v

) = µ(u,v), as opposed to ϕ(

u,

$$\bar{\mu}$$

v

) = µϕ(

u,v

). For example, the natural inner product to deal with functions

f

: ℝ

→

$$\mathbb{C} $$

, especially Fourier series, is

$$\langle{f,g}\rangle = \int\limits_{\pi}^{-\pi}f(x)\overline{g(x)}dx,$$

which is semilinear (but not linear) in g.