Computing Eigenvectors

How to Compute Eigenvalues and Eigenvectors of a Matrix

How do you find Eigen-axes?

We have two options.

1) (Computationally Hard)

A \vec{v} = \lambda \vec{v}.

2) (Computationally Easy)

\left( A - \lambda I \right) \vec{v} = \vec{0}.

Of course, we are going to go with the easier option.

Understanding the options and the Difference Matrix

You already understand option 1. So how does Option 2 work? $\lambda I$ is a matrix with $\lambda$ s on the diagonal, so it multiplies each of $\vec{v}$ 's componentsby $\lambda$ :

\left( \lambda I \right) \vec{v} = \begin{bmatrix} \lambda & 0 \\ 0 & \lambda \end{bmatrix} \vec{v} = \begin{bmatrix} \lambda x \\ \lambda y \end{bmatrix} = \lambda \vec{v}.

In order to find an eigen-axis, we need to find vectors for which the matrix $A$ and $\lambda I$ have the same effect. Two things have the same effect when there is no difference between their effects. So we just look for that! This yields the magic formula for our second option:

A\vec{v} = \lambda \vec{v} \iff A \vec{v} - \lambda I \vec{v} = \vec{0} \iff (A - \lambda I)\vec{v} = \vec{0}.

So to find an entire eigen-axis for an eigenvalue $\lambda$ , we look for vectors where the difference between $A \vec{v}$ and $\lambda \vec{v}$ is $\vec{0}$ . The Difference Matrix is $(A-\lambda I)$ , the matrix that captures this difference.

Problem 1: Find an Eigenvector with Zero Difference

Below is a linear transformation given by an unknown 2x2 matrix. Displayed are $\vec{v}$ , $A \vec{v}$ , $-\lambda I \vec{v}$ , and finally the difference $(A-\lambda I)\vec{v})$ . Move $\vec{v}$ around to make the purple difference vector $(A-\lambda I)\vec{v})$ equal to $\vec{0}$ . The result is an eigenvector with an eigenvalue of 2.

Enter the eigenvector you found here.

Problem 2: Finding the Other Eigenvalue

You should see a slider now that lets you control the value for $\lambda$ . Move $\vec{v}$ around until you find the other eigen-axis, then move the slider until the difference vector disappears. What second value for $\lambda$ did you find?

Enter the second eigenvalue you found below.