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42 Introduction to Control Theory

EXAMPLE 2.5–4: Equivalent Vector Norms

1. It can be shown that for

2. Consider again the vector of Example 2.5.3. Then we can check that

Matrix Norms
In systems applications, a particular vector x may be operated on by a matrix
A to obtain another vector y=Ax. In order to relate the sizes of x and Ax we
define the induced matrix norm as follows.

DEFINITION 2.5–4 Let ||x|| be a given norm of x∈ . Then each m×n matrix
A, has an induced norm defined by

where sup stands for the supremum.
It is always imperative to check that the proposed norms verify the
conditions of Definition 2.5.3. The newly defined matrix norm may also be
shown to satisfy

||AB|| i ≤ ||A|| i ||B|| i
for all n×m matrices A and all m×p matrices B.

Copyright © 2004 by Marcel Dekker, Inc.