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8.8.1 Finding the Eigenvalues of a Matrix
To find the eigenvalues, note that the above definition of eigenvectors and
eigenvalues can be rewritten in the following form:

(M − λ ) I v = 0 (8.28)

where I is the identity n ⊗ n matrix. The above set of homogeneous equations
admits a solution only if the determinant of the matrix multiplying the vector
v is zero. Therefore, the eigenvalues are the roots of the polynomial p(λ),
defined as follows:

p( )λ = det(M − λI ) (8.29)

This equation is called the characteristic equation of the matrix M. It is of
degree n in λ. (This last assertion can be proven by noting that the contribu-
tion to the determinant of (M – λI), coming from the product of the diagonal
n
elements of this matrix, contributes a factor of λ to the expression of the
determinant.)

Example 8.9
Find the eigenvalues and the eigenvectors of the matrix M, defined as follows:

 2  4
M =  
12/  3

Solution: The characteristic polynomial for this matrix is given by:

p()λ = ( −2 λ )( −3 λ ) ( )( / )− 4 1 2 = λ − 5 λ + 4
2

The roots of this polynomial (i.e., the eigenvalues of the matrix) are,
respectively,

λ = 1 and λ = 4
1
2
To find the eigenvectors corresponding to the above eigenvalues, which we
shall denote respectively by v and v , we must satisfy the following two
1 2
equations separately:

 2 4   a   a
  
  12/ 3   b = 1  
  b
and

© 2001 by CRC Press LLC