  # Eigenspace

## What is an Eigenspace?

An eigenspace is the collection of eigenvectors associated with each eigenvalue for the linear transformation applied to the eigenvector. The linear transformation is often a square matrix (a matrix that has the same number of columns as it does rows). Determining the eigenspace requires solving for the eigenvalues first as follows:

Equation 1

(A - λI)x = 0

Equation 2

det(A - λI) = 0

Equation 3

A - λI

Where A is the square matrix, λ is the eigenvalue, I is the identity matrix and x is the eigenvector. Equation 1 is the equation we are trying to solve that will give us the eigenvalues and eigenvectors. Equation 2 is the determinant of the matrix (A - λI) and is used to solve for the eigenvalues. There can be as many eigenvalues as there are rows in the matrix A. Once the eigenvalues are calculated, use them in Equation 3 to determine the eigenvectors. Plug in each eigenvalue and calculate the matrix that is Equation 3. Reduce or normalize the elements of the matrix and the eigenspace can be extracted from there. Knowing the eigenspace provides all possible eigenvectors for each eigenvalue.

### Practical Uses of an Eigenspace

• Image processing

– Eigenvectors, or eigenfaces, are used to express the brightness of each pixel in the image of a face for use in the facial recognition branch of biometrics.

• Geology – The eigenspace can be used to summarize in 3-D space the orientation of the clast of glacial till.
• Vibration analysis – Eigenspace describes the shapes of the vibration modes of an object for each eigenvalue or natural frequency, referred to in this context as an eigenfrequency.