Eigen Values and Vectors
- Given a matrix, M, the scalar λ ≠ 0 is an Eigen-value of M, and the vector v is a corresponding Eigen-vector of M, if
- M v = λ v.
- The direction of v, is significant, not its magnitude, so v is usually normalized, ||v|| = 1. An Eigen-vector is a "fixed-point" 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,
- λ2 - (a + d)λ + (ad - bc) = 0,
- λ = {(a + d) ± √((a + d)2 - 4(ad - bc))} / 2
- = {(a + d) ± √((a - d)2 + 4bc)} / 2,
- (λ - a)(λ - d) - bc = 0,
- 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 - x + 2y = 1 y,
- or
- 3x + 2y = 4x,
- x + 2y = 4y,
- x = 2y, e.g., (2, 1).
3 2 1 2 2 1 = 4 2 1 - x + 2y = 4y,
- In general an n×n matrix may have
up to n Eigen-values, not necessarily distinct,
and some or all may be complex.
- A real symmetric n×n matrix has n real Eigen-values.
- The Eigen-values of a diagonal matrix are just the diagonal elements.
- The Jacobi algorithm is an algorithm to find the Eigen-values and Eigen-vectors of a symmetric matrix.
- A real symmetric n×n matrix has n real Eigen-values.