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,

> [ [ 2 , 1 ],

> [ -1 , 4 ] ] );

> eigenvalues( A );

>

> eigenvectors( A );

>

Case 3: Repeated Eigenvalue

> lambda := 3 ;

> V1 := vector( 2, [ 1 , 1 ] );

> V2 := vector( 2, [ x2 , y2 ] );

>

The straight-line solution is:

> Y1 := exp( lambda *t) * convert( V1, matrix );

>

To find a second linearly independent solution we must find any vector that satisfies or, equivalently, .

> sys := geneqns( evalm( A-lambda*I2 ), V2, V1 );

> soln := solve( sys, { x2, y2 } );

> V2 := subs( soln , x2=0, y2=0, evalm(V2) );

>

> Y2 := exp( lambda *t) * ( t * convert( evalm( V1 ), matrix ) + convert( evalm( V2 ) , matrix ) );

>

> SOLgen := c1 * Y1 + c2 * Y2;

>

>