Analytic Solution for Linear Homogeneous Systems (Eigenvalue Analysis)

> with( linalg ):

Warning, new definition for adjoint

Warning, new definition for norm

Warning, new definition for trace

> I2 := matrix( 2, 2, [[1,0],[0,1]] ); # 2x2 identity matrix

> A := matrix( 2, 2,

> [ [ 1 , 5 ],

> [ -2 , -1 ] ] );

> eigenvalues( A );

>

> eigenvectors( A );

>

Case 2: Complex Eigenvalues

> alpha := 0 :

> beta := 3 :

> lambda := alpha + I * beta;

> V := vector( 2, [ -1/2-3/2*I , -1 ] );

>

The linearly independent solutions are found by the usual steps: exp( eigenvalue * t ) * eigenvector, but recall that the two solutions are the real and imaginary parts of this expression. These calculations are messy - by hand. With Maple it's a little better, but there are some extra commands that must be used.

We know that the two solutions will consist of the product of an exponential formed with the real part of the eigenvalues and a vector with trigonometric terms formed using the imaginary part of the eigenvalues. Let's begin by looking at the part of the solution that leads to complex-valued expressions:

> assume( t, real );

> mess := exp( I* beta *t ) * evalm( V );

> mess2 := convert( evalc( evalm( mess ) ), trig );

> t := 't';

> messRE := map( Re, mess2 );

> messIM := map( Im, mess2 );

>

Using the real and imaginary parts we can construct the two linearly independent solutions for this system:

> Y1 := exp( alpha *t) * convert( evalm( messRE ), matrix );

> Y2 := exp( alpha *t) * convert( evalm( messIM ), matrix );

>