7.2 EIGENVALUES

Equation (7.1) may be arranged to the form

Here, the matrix

is called the characteristic matrix of the matrix A. For the homogeneous set of equations represented ny eq (7.3), Section 5.8 shows that nontrivial solutions can only be obtained if

The determinant of K is called the characteristic determinant or the characteristic function of the matrix A, and it is clearly a polynomial in ?

This polynomial is called the characteristic polynomial of the matrix A and when it set equal to zero in accordance with eq (7.5),

it is called the characteristic equation of the matrix A. The ?'s are the roots of p( ?) = 0, and because this equation is called the characteristic equation, the roots are called the characteristic values or values or eigenvalues. The eigenvalues may be real or complex and, of course, some eigenvalues may be repeated.

Example 7.1

Consider the matrix

and form the characteristic matrix, K=[A ? ?I],

The evaluation of the determinant of the characteristic matrix yields the characteristic polynomial and then the characteristic equation

or

The roots of p( ?)=0 are easy to find and these are the eigenvalues that may be listed in order, with the largest first,

and

These eigenvalues are real and unequal (distinct).