The reason to find the eigenvectors is to diagonalize the matrix, so after
finding the eigenvectors, we'll go ahead and diagonalize the matrix A so we
can check to see that they are the correct eigenvectors.



We are given the eigenvalues. For 











So k=20.

We could do the same with the other eigenvector, but it will also give k=20.

So now we have



To diagonalize A we want to find matrices D and S so that we can write A as
 


where D is the diagonal matrix with the two eigenvalues on the 
main diagonal:



and the matrix S is 



where the V's are the two column eigenvectors for the two eigenvalues

We find V1 which is the eigenvector for the eigenvalue λ=-4.

We find solutions for










Divide thru by -18


We can take x1=1 and x2=-5

So 



Now we do the same for the other eigenvalue

---

We find solutions for











We can take x1=1 and x2=4

So 



Now we have done that was asked for, for we have the two eigenvectors. So we
can stop here.  





-------------------------------------

But let's check to make sure this matrix whose columns are the two
eigenvectors:



provide the correct diagonalization of A.

Since the determinant of S is 9, to find S-1 we only need to swap the 
elements on the the main diagonal and change the signs of the other two
elements, then multiply by 1/9:



Then if we've done everything right, this will be true:



So we check to see if this is the correct diagonalization of A:















So the eigenvalues are correct because 



is the correct diagonalization of A.


Edwin