Created Friday 15 November 2013
- http://tutorial.math.lamar.edu/Classes/DE/LA_Eigen.aspx
- http://www.sosmath.com/matrix/eigen1/eigen1.html
Let `vec v` be a vector, `alpha` a scalar, and M be a matrix.
`alpha` is an eigenvalue of M and `vec v` is an eigenvector of M if `M vec v = alpha vec v`.
- Matrix multiplication of a matrix with its eigenvectors is the same as scalar multiplication.
Calculating the Eigenvalues and Eigenvectors
If M is an n x n matrix then `det(M - alpha I)` is an nth-degree polynomial (in `alpha`) called the characteristic polynomial.
To find the eigen values of a matrix, all you need to do is find the roots of the characteristic polynomial.
Example
`M = ((2,7), (-1,-6))`
Then
`M - alpha I`
`= M - ((alpha,0), (0, alpha))`
`= ((2-alpha,7), (-1,-6-alpha))`
So
`det(M - alpha I)`
`= (2-alpha)(-6-alpha) - (-7)`
`= -12+4alpha+alpha^2 + 7`
`= alpha^2 + 4 alpha - 5`
`= (alpha + 5)(alpha - 1)`
The eigenvalues are -5 and 1.