Computing Eigenvectors

How to Compute Eigenvalues and Eigenvectors of a Matrix

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How do you find Eigen-axes?

We have two options.

1) (Computationally Hard)

Av⃗=λv⃗.A\vec{v}=\lambda\vec{v}.

2) (Computationally Easy)

(A−λI)v⃗=0⃗.\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? λI\lambda I is a matrix with λ\lambdas on the diagonal, so it multiplies each of v⃗\vec{v}'s components by λ\lambda:

(λI)v⃗=[λ00λ]v⃗=[λxλy]=λv⃗.\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 AA and λI\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:

Av⃗=λv⃗  ⟺  Av⃗−λIv⃗=0⃗  ⟺  (A−λI)v⃗=0⃗.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 λ\lambda, we look for vectors where the difference between Av⃗A\vec{v} and λv⃗\lambda\vec{v} is 0⃗\vec{0}. The Difference Matrix is (A−λI)(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 2×22\times2 matrix. Displayed are v⃗\vec{v}, Av⃗A\vec{v}, −λIv⃗-\lambda I\vec{v}, and the difference (A−λI)v⃗(A-\lambda I)\vec{v}. Move v⃗\vec{v} around to make the purple difference vector equal to 0⃗\vec{0}. The result is an eigenvector with eigenvalue λ=2\lambda=2.

Enter the eigenvector you found here.

Answer Problem 1 correctly to reveal Problem 2.