Obtaining the Eigenvectors from a Matrix

The Eigenvector of a matrix can be found by first computing the eigenvalues (using the characteristic equation). Then, find the solution space of the linear system: $(\mathbf{A}-{\lambda }_i\mathbf{I})\vec{v}=0$ for each eigenvalue, ${\lambda }_i$.

1. Compute the eigenvalues §

This, then can be expressed as a polynomial:

As such, the roots of the characteristic polynomial would be ${\lambda }_1,{\lambda }_2,\dots {\lambda }_i$

2: Find the solution space of the associated linear system §

Now, for a chosen eigenvalue, $\lambda$ , the associated homogenous linear system is given by:

$$(\mathbf{A}-\lambda \mathbf{I})\vec{v}=0$$

3: Convert to an appropriate basis (Optional) §

Once the solution space has been found, there may be eigenvectors that are linearly dependent. As such, you can reduce the space to a basis:

Alternatively:

This is useful when you need to determine the geometric multiplicity of a given eigenvalue. Let $E_{\lambda }$ be the eigenspace of the eigenvalue $\lambda$. Also, let the basis we have found be ${\mathcal{B}}_{\lambda }$ Then:

$$\text{Geometric multiplicity of }{E}_{\lambda }=\text{dim}(E_{\lambda })=\text{No. of elements in }{\mathcal{B}}_{\lambda }$$