# Eigenvalues & Eigenvectors

For a square matrix , a scalar is an eigenvalue of , where:

This can be rewritten as a homogenous equation in an unknown vector :

This equation has infinitely many non-trivial solutions for , where:

This is the characteristic equation of , and the eigenvalues are scalars that satisfy this. Since the characteristic equation is an -th degree polynomial, an matrix will have eigenvalues for .

Corresponding to each eigenvalue , eigenvectors are non-trivial solutions of:

## Example

Eigenvalues and vectors of:

The characteristic equation and it's solutions:

Eigenvector for :

Eigenvector for :

## Spectral Decomposition

An x matrix has eigenvectors and associated eigenvectors .

is an x matrix of column eigenvectors, and is an x diagonal matrix of eigenvalues

for all matrices

In general, eigenvectors of are linearly independent and so exists. The spectral decomposition of a matrix can then be written:

This allows for diagonalisation of a matrix in terms of its eigenvectors, and for breaking down a multi-dimensional problem into a set of single dimensional problems.

• This is only possible if all eigenvectors are linearly independent.
• If any are repeated then this is not the case

If is a symmetric matrix, then the eigenvectors are mutually orthogonal, ie for all . If these eigenvectors are orthonormalised (of unit length), then the matrix of eigenvectors is an orthogal matrix, meaning its transpose is equal to it's inverse. Hence, the spectral resolution of a symmetric matrix is:

### Example

Find the spectral resolution of, and hence diagonalise:

The eigenvalues of are and . These can then be used to compute the corresponding eigenvectors:

Using :

The spectral resolution of is given by:

can then be diagonalised by :