Maths - Matrix algebra - Eigenvectors and Eigenvalues

These quantities can have a geometric meaning and are also useful in matrix algebra, the geometric meaning is discussed on this page, it tells us something about the symmetry of a transform.

An eigenvector is a vector whose direction is not changed by the transform, it may be streached, but it still points in the same direction.

Each eigenvector has a corresponding eigenvalue which gives the scaling factor by which the transform scales the eigenvector. So the eigenvector is a vector and the eigenvalue is a scaler.

A given transform may have more than one eigenvector and eigenvalue pair depending on how many dimensions we are working in. For instance:

• If we are working in 2 dimensions there are upto 2 eigenvector and eigenvalue pairs.
• If we are working in 3 dimensions there are upto 3 eigenvector and eigenvalue pairs.

and so on.

As an example, if we have a rotation transform in 3 dimensions, then the eigenvector would be the axis of rotation since this is not altered by the transform and the corresponding eigenvalue would be +1 since the axis is not scaled by the rotation. If we have a rotation in 2 dimensions then the eigenvectors would be ±i where i is √-1 since all vectors in the plane change direction.

Eigenvalues

The eigenvalues of a matrix [M] are the values of such that:

[M] {v} = {v}

where {v} = a vector

this gives:

|M - I| = 0

where I = identity matrix

this gives:

so

(m00- ) (m11- ) (m22- ) + m01 m12 m20 + m02 m10 m21 - (m00- ) m12 m21 - m01 m10 (m22- ) - m02 (m11- ) m20 = 0

the values of λ are the eigenvalues of the matrix

Eigenvectors

Associated with each eigenvalue λ_i is an eigenvector {u_i} such that:

[M] {u_i} = λ_i {u_i}

where:

• [M] is a matrix
• λ_i is its eigenvalues (i=1,2,3)
• {u_i}is its eigenvectors

Linear Functions