Eigen Values and Vectors

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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.
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