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 2 4   c   c
    = 4  
 12/ 3   d   d

From the first set of equations, we deduce that: b = –a/4; and from the second
set of equations that d = c/2, thus giving for the eigenvectors v and v , the
1 2
following expressions:

 − 1 
v = a  
1 14
 /

 − 1 
v = c  
2  − /
12
It is common to give the eigenvectors in the normalized form (that is, fix a and
c to make vv = v v = 1, thus giving for v and v , the normalized
1 1 2 2 1 2
values:

16  − 1   − .
0 9701
v =   =  
0 2425 
1 17 14  .
 /
0 8944
4  − 1   − .
v =   =  

12
0 4472
2
5 − /  − .
8.8.2 Finding the Eigenvalues and Eigenvectors Using MATLAB
Given a matrix M, the MATLAB command to find the eigenvectors and
eigenvalues is given by [V,D]=eig(M); the columns of V are the eigen-
vectors and D is a diagonal matrix whose elements are the eigenvalues. Enter-
ing the matrix M and the eigensystem commands gives:

V =
-0.9701 -0.8944
-0.2425 -0.4472
D =
1 0
0 4
Finding the matrices V and D is referred to as diagonalizing the matrix M. It
should be noted that this is not always possible. For example, the matrix is
not diagonalizable when one or more of the roots of the characteristic poly-

© 2001 by CRC Press LLC