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.