Start Maple from Start Menu/Program Files or using the icon in f:\math115

Load the linear algebra progams with the command:
>    with(linalg):

Complete the previous week's lab questions if you haven't done them
before now. Now try these eigen-questions:

Predict and then calculate the eigenvalues and eigenvectors of this matrix:
>    G:=matrix(5,5,[4,5,1,-2,0,0,3,-2,2,1,0,0,7,1,-2,0,0,0,4,-3,0,0,0,0,-3]);

Check your answer with 
>   eigenvals(G);
and 
>   eigenvects(G);
reading carefully the help pages which describe their format


Given H:=matrix([[-1, 3, -3], [5, -3, 6], [6, -6, 9]]);
check that although H has rank 2 it has 3 different eigenvalues
by using determinant operations

Show that
>  J:=matrix([[-2, 6, 2], [-2, 5, 1], [-2, 4, 2]]);
has rank 3 but only two eigenvectors, again using determinants

Given this matrix
>   K:=matrix([[115, 198, 87, 81], [-126, -218, -96, -90], [72, 126, 55, 54], [66, 114, 51, 46]]);
Verify that 
> v1:=matrix(4,1,[-3, 0, 3, 1]);
and 
> v2:=matrix(4,1,[1, 0, -6, 5]);
are eigenvectors and deduce their eigenvalues.

Use determinant row and column operations to find the remaining eigenvalues of K, using
what you have already learned to help you isolate the factors you need.


Now evaluate H^6 directly, and then form a matrix of eigenvectors, P, and a diagonal matrix of eigenvalues E.
Calculate the product of P, E^6 and inverse(P) and verify it is the same as H^6.
Check that you get the same answer if you switch around the order of the eigenvalues in the diagonal matrix.

What are the eigenvalues of H^6 ? 
Try to predict the eigenvalues and eigenvectors of J^6 and K^6 and check your answer.

What are the eigenvalues of H+J ? HJ ? JH ? 2J ? kJ ? 
which of these are true relations and which are coincidences?