Geometrical interpretation of eigenvectors and eigenvalues

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°).