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48 Introduction to Control Theory

This concludes our brief review of norms as they will be used in this book.
Inner Products

An inner product is an operation between two vectors of a vector space which
will allow us to define geometric concepts such as orthogonality and Fourier
series, etc. The following defines an inner product.

DEFINITION 2.5–8 An inner product defined over a vector space V is a
function <.,.> defined from V to F where F is either or such that x, y,
z, ∈V

1. <x, y>=<y, x>* where the <.,.>* denotes the complex conjugate.
2. <x, y+z>=<x, y>+<x, z>
3. ,

4. <x, x>≥0 where the 0 occurs only for x=0 V

EXAMPLE 2.5–9: Inner Products

n
The usual dot product in is an inner product.

We can define a norm for any inner product by

(2.5.2)

Therefore a norm is a more general concept: A vector space may have a norm
associated with it but not an inner product. The reverse however is not true.
Now, with the norm defined from the inner product, a complete vector space
in this norm (i.e. one in which every Cauchy sequence converges) is known as
a Hilbert Space

Matrix Properties

Some matrix properties play an important role in the study of the stability of
dynamical systems. The properties needed in this book are collected in this
section. We will assume that the readers are familiar with elementary

Copyright © 2004 by Marcel Dekker, Inc.