Page 69 -

50 Introduction to Control Theory

THEOREM 2.5–2: Rayleigh-Ritz Let A be a real, symmetric n×n positive-
definite matrix. Let min be the minimum eigenvalue and max be the maximum
eigenvalue of A. Then, for any ,

THEOREM 2.5–3: Gerschgorin Let A=[a ij] be a symmetric n×n real matrix.
Suppose that

If all the diagonal elements are positive, i.e. a ii >0, then the matrix A is positive
definite.

EXAMPLE 2.5–10: Positive Definite Matrics

Consider the matrix

(1)

Its symmetric part is given by

(2)

This matrix is positive-definite since its eigenvalues are both positive (1.8377,
8.1623). Of course, Gershgorin’s theorem could have been used since the
diagonal elements of A s are all positive and

T
On the other hand, consider a vector x=[x 1 x 2 ] and its 2-norm, then

Copyright © 2004 by Marcel Dekker, Inc.