We will first discuss eigenvectors and eigenvalues using conventional matrix notation although eigenvectors and eigenvalues are not specific to matrices and other algebras and notations can be used.
The eigenvalues of a matrix [M] are the values of a vector 'v' such that:
[M] v = λ v
where:
- v = eigenvector
- λ = lambda = eigenvalue
In other words if we treat the matrix 'M' as a transform which vectors are not changed (or are only scaled) by the matrix.
Examples
| matrix | vectors not changed |
|---|---|
| rotation | centre of rotation (often origin), for 2 dimensional case, or axis of rotation for higher dimensional case. |
| translation | vectors along the direction of movement |
| scaling | all |
| inertia tensor | principal moments of inertia |
One of the reasons that eigenvectors are so important is that the points that do not move are what defines the symmetry of a given operation
So far the assumption is that the matrix contains real values, if the matrix is over complex numbers for example, then these results may be modified (a common geometric interpretation of the imaginary operator 'i' is rotation by 90°).
