TABLE OF CONTENTS In this section the concept of eigendecomposition, eigenvectors, and eigenvalues will be introduced. This concept will be used in later sections to estimate frequencies.
If A is a given square matrix, a constant ? and a vector X can be found such that
| (14.78) |  |
where ? is called an eigenvalue and X is called its corresponding eigenvector. This process is called the eigende composition of A. To find ? and X, the above equation can be written as
| (14.79) |  |
where I is the identity matrix. In order to have a nontrivial solution (i.e., X ? 0), the determinant of A - ?I should equal to zero. For example, if
| (14.80) |  |
The eigenvalues solved are ? 1 = -0.3723 and ? 2 = 5.3723, which can be found using the MATLAB "eig" command. With each eigenvalue there is an eigenvector. The corresponding eigenvector X i = [ x i 1 x i 2] T where superscript T representing the transpose of a matrix can be solved from (14.78). The eigenvectors are found from MATLAB as X 1 = [-.8246 .5658] T and X 2 = [-.4160 -.9094] T with the restriction x i 1 2 + x i 2 2 = 1, i = 1, 2.
Let us use a simple example to demonstrate the application of eigenvectors and...
