The notion of eigenvalue and eigenvector

Eigenvalues and eigenvectors: Eigenvalues and eigenvectors

The notion of eigenvalue and eigenvector

We start with an example of a linear mapping in the plane. Change the vector $\vec{v}$ by dragging its arrowhead and see what the effect is on $L(\vec{v})$ .

Below we look at a geometric plane with a fixed origin $O$ . Drawn in it is a green position vector $\vec{v}$ , which you can change by dragging the tip at the end of the arrow.
We consider a linear mapping $L$ on this plane and always show in red the image $L(\vec{v})$ of $\vec{v})$ under this mapping.

Change $\vec{v}$ and watch the image change $L(\vec{v})$ simultaneously.
In this example there are two situations to create: existence of a vector

$\vec{u}$ with $L(\vec{u})=-\vec{u}$ .
$\vec{w}$ with $L(\vec{w})=2\vec{w}$ .

Try to find such vectors by a trial-and-improvement method of dragging $\vec{v}$ .

0,0

–o+←↓↑→

$\vec{v}$

$L(\vec{v})$

New example

If $L$ is a linear transformation of a vector space, we call a nonzero vector $\vec{v}$ with the property that $L(\vec{v})=\lambda \vec{v}$ for some scalar $\lambda$ an eigenvector of $L$ and $\lambda$ the eigenvalue corresponding to the eigenvector.

In other words, an eigenvector of the linear transformation is mapped onto a multiple of itself, and the multiplication factor is called the corresponding eigenvalue. If $\vec{v}$ is an eigenvector with corresponding eigenvalue $\lambda$ , then every scalar multiple of $\vec{v}$ is that too. If $\vec{u}$ and $\vec{v}$ are eigenvectors corresponding to the same eigenvalue $\lambda$ , then the sum vector $\vec{u}+\vec{v}$ is that too. In fact, the set of eigenvectors corresponding to a single eigenvalue is a vector space, which is a subspace of the covering vector space we work in. This is called the eigenspace of the eigenvalue.