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,

 

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 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.