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