# Eigenvectors

Unit Vectors:

Matrix:

Eigenvector 1: [1,0]

Eigenvector 2: [0,1]

Eigenvector 2: [0,1]

Unit Vectors:

Matrix:

Eigenvector 1: [1,0]

Eigenvector 2: [0,1]

Eigenvector 2: [0,1]

A vector \( \bf{v} \) is an eigenvector of a matrix \(M\) (with eigenvalue \(\lambda\) if
\[ M\bf{v} = \lambda\bf{v} \]
For \( 2 \times 2\) matrices, eigenvectors can be visualized graphically: these are the vectors whose direction remains unchanged when they are multiplied by \(M\) (note their magnitude will change by a factor of \(\lambda\)).

This applet demonstrates the effect of a matrix being applied to a variety of unit vectors. As you change the matrix entries, you'll see blue lines appear to radiate from each unit vector - the end of these line segments is the point \( M\bf{u} \) for each unit vector \(\bf{u}\). If the matrix has real-valued eigenvectors, they'll be calculated and shown in red. If a unit vector is close to an eigenvector, then the blue line will point in nearly the same direction as the unit vector (this is also true for the unit vector pointing in the opposite direction). If a real eigenvector does not exist, then there are no unit vectors \(\bf{u}\) that point in the same direction as \( M\bf{u} \).

This applet was created using JavaScript and the Raphael library. If you are unable to see the applet, make sure you have JavaScript enabled in your browser. This applet may not be supported by older browsers.