Definition.  If A is an n x n  matrix, then a nonzero vector x in Rn is called an eigenvector of A if Ax is a scalar multiple of x;  in other words,

for some scalar .  The scalar is called an eigenvalue of A and x is the eigenvector corresponding to .

Theorem:  If A is an n x n matrix, then the following are equivalent.

a)  is an eigenvalue of A.

b)  The system of equations (I - A)x = 0 has nontrivial solutions.

c)  There is a nonzero vector x in Rn such that Ax = x.

d)   is a real solution of the characteristic equation det(I - A) = 0.