As usual start Maple and load all the linear algebra progams.

The matrix underlying the recurrence c(n+1)= 2*c(n) + 115*c(n-1) + 364*c(n-2) is
> C:=<<2,1,0>|<115,0,1>|<364,0,0>>;

Diagonalise C utilising
> (v,P):=Eigenvectors(C);
and notice that its eigenvectors have the special "powers
of eigenvalues" formula they are supposed to have.

Form Ck as the kth power of C by diagonalisation and now suppose that the 
first three values in the sequence are c(0)=10, c(1)=7 and c(2)=-17. 
Multiply Ck by the initial values vector, remembering to use the correct order.

Calculate c(3) and c(4) using the recurrence then use the power matrix to
get c(k) and check with your values with k=3 and 4.

Note that Maple can find the solution in one step:
> rsolve({c(n+1)= 2*c(n) + 115*c(n-1) + 364*c(n-2),c(0)=10, c(1)=7, c(2)=-17},c(k));

Find initial values for the sequence which would give a solution with one or 
two different eigenvalue powers in the final solution for c(k). 
Determine a link between these initial values and the eigenvectors of C.




Create a 4x4 triangular matrix E with an eigenvalue of multiplicity 3 as
described in class and ensure it has only two eigenvectors of 
the eigenvalue of multiplicity 3.

Now create a random invertible 4x4 matrix Q and verify that this 
matrix B has the same eigenvalues and multiplicities, but more 
complicated eigenvectors than E:
> F:=Multiply(Multiply(Q,E),MatrixInverse(Q));

Now create a 4x4 matrix G with only two eigenvalues of multiplicity 2 
and two eigenvectors and ensure that G has no zeros in its entries.



Let J be the two by two matrix underlying the twin recurrences:
a(n+1)= 7*a(n) + 12*b(n) and b(n+1)= 8*a(n) + 3*b(n)

If a(0)=5 and b(0)=7, determine a(1), a(2) and a(3) and b(1), b(2) and b(3) by
using (for k=1, 2, and 3)
> Multiply(MatrixPower(J,k),<5,7>);
What ratio are the coordinates approaching?

Pick another random pair of values for a(0) and b(0) and verify that 
the solutions approach the same ratio.

Now pick b(0) = - a(0) for some value and see what the solutions are for a(n) and b(n)


Create a matrix M which has two eigenvalues of absolute value less than 1 and verify 
that no matter what the initial values p and q are the values get smaller and smaller.
> Multiply(MatrixPower(M,k),<p,q>);