MATLAB.5
                                                                   
Linear Systems

Consider

 X'=AX
 
where A is nxn. If there are n linearly independent eigenvectors
for A , {v(1),...,v(n)} , then a fundamental set of solutions is given by
{exp(s(i)*t)v(i),i=1..n}. If n is large though this may be hard to find by hand.
Here we show how to use Matlab to find the eigenvalues and eigenvectors.

example

Let
A =

    -1     0     2
     2     3     6
    -2     0    -1

Type :

***********************
[E,V]=eig(A)
***********************
We get

E =
  
        0            0.5000                  0.5000         
      1.0000      0.1000- 0.7000i    0.1000+ 0.7000i
        0           0+ 0.5000i            0- 0.5000i


V =

   3.0000                      0                        0         
        0           -1.0000+ 2.0000i                    0         
        0                     0                 -1.0000- 2.0000i

The diagonal matrix V lists the eigenvalues of A and the
corresponding columns of E are the eigenvectors. Using this information
we can write down 3 linearly independent real solutions ( a fundamental set ).

X(1)=exp(3*t)*[0,1,0]'
X(2)=exp(-t)*[0.5*cos(2*t),0.1*cos(2*t)+sin(2*t)*0.7,-0.5*sin(2*t)]'
X(3)=exp(-t)*[0.5*sin(2*t),-0.7*cos(2*t)+0.1*sin(2*t),0.5*cos(2*t)]'
Note v' is the transpose of v.

What do you do if there are not n linearly independent  eigenvectors? To see 
how to complete a fundamental set in this case look at Polking,  pages 163-170..



ASSIGNMENT  5 :
Write down a real fundamental set if
1.

A =

  -0.2       0        0.2
   0.2      -0.4      0
   0         0.4     -0.2
        
2.
                                                                   
A =

     4     1     1     7
     1     4    10     1
     1    10     4     1
     7     1     1     4