An eigenvector of a linear transformation is a vector (non-zero) that, when the linear transformation is applied to it, changes by only a scalar factor. This scalar factor is called eigenvalue.

Where:

• $A$: Transformation Matrix
• $\overrightarrow{v}$: Eigenvector
• $\lambda$: Eigenvalue

To calculate the Eigenvectors we need to find the Eigenvalues first.

Finding the Eigenvalues

From

follows

which is only solvable if $det(A - \lambda I) = 0$ (characteristic equation).

This means $(A - \lambda I)$ not invertible.

The solution of the characteristic equation ($det(A - \lambda I) = 0$) are the eigenvalues.

Eigenvectors

There is one independent Eigenvector for each Eigenvalue.

For each Eigenvalue solve

to get the corresponding Eigenvector $\overrightarrow{v}$ for a given Eigenvalue $\lambda$.