Definition:
If A is a nxn matrix, L a scalar is an eigenvalue of A, it there
is a NON-ZERO vector X so that AX = LX. (Usually L is the greek
letter lambda.)

This means 0 is an eigenvalue <=> homogenuous equation AX = 0
has a non-zero solution <=> AX = 0 has oo-many solutions. That
is rref(A) has a column with no pivot.

If we know L is eigenvalue then B = A-LI has non-zero X so that BX = 0.
(because BX = (A-LI)X = AX - LIX = LX - LX = 0.) So again, finding
eigenvectors is the same as finding non-zero solutions to homogenuous
equations with oo-many solutons.

When the rref matrix has two non-pivot columns we found `fundatmental
solutions', two independent vectors by first making the first non-pivot 
variable 1 and the second 0, and second by making the first non-pivot
variable 0 and the second 1.

The characteristic polynomial of the nxn matrix A is p(s) = det(A-sI)

The scalar r is an eigenvalue for the nxn matrix A <=> r is a root
of the characteristic polynomial p(s), that is p(r) = 0.

Complex numbers can be eigenvectors. The matrices in this class are 
carefully chosen so the only eigenvalues are real. For example, symmetric
matrices have only real eigenvalues.

If X1 and X2 are eigenvectors so that A*X1 = L1*X1 and A*X2 = L2*X2 then
for any X = C1*X1 + C2*X2 you can compute A*X by
    A*X = A(C1*X1 + C2*X2) = C1*A*X1 + C2*A*X2 = C1*L1*X1 + C2*L2*X2

We did deep background showing for every lambda, d/dx has lambda
as an eigenvalue with eigenfuction exp(lambda x). The equation
AX = lambda X became the ODE y' = lambda y in this case. 

We also say the sin(x) and cos(x) solutions of (D^2+I)y = 0
had stuff in common with finding two independent solutions of AX = 0.