Eigen Values and Vectors
 Given a matrix, M, the scalar λ ≠ 0 is an Eigenvalue of M, and the vector v is a corresponding Eigenvector of M, if
 M v = λ v.
 The direction of v, is significant, not its magnitude, so v is usually normalized, v = 1. An Eigenvector is a "fixedpoint" of M in direction, but not in magnitude in general.
 For example,

a b c d x y = λ x y  ax + by = λx
 cx + dy = λy
 x(λ  a) = by
 y(λ  d) = cx
 x(λ  a) = by = bc x / (λ  d),
 (λ  a)(λ  d)  bc = 0,
 λ^{2}  (a + d)λ + (ad  bc) = 0,
 λ = {(a + d) ± √((a + d)^{2}  4(ad  bc))} / 2
 = {(a + d) ± √((a  d)^{2} + 4bc)} / 2,
 e.g., a = 3, b = 2, c = 1, d = 2,
 λ = {5 ± √(1 + 8)} / 2 = {5 ± 3} / 2 = 1, or 4,
 giving either
 3x + 2y = 1 x,
 x + 2y = 1 y,
 x =  y, e.g., (1, 1)

3 2 1 2 1 1 = 1 1 1  or
 3x + 2y = 4x,
 x + 2y = 4y,
 x = 2y, e.g., (2, 1).

3 2 1 2 2 1 = 4 2 1  In general an n×n matrix may have up to n Eigenvalues, not necessarily distinct, and some or all may be complex.
 A real symmetric n×n matrix has n real Eigenvalues.
 The Eigenvalues of a diagonal matrix are just the diagonal elements.
 The Jacobi algorithm is an algorithm to find the Eigenvalues and Eigenvectors of a symmetric matrix.