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 \(\vec{v}\).

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.