Computing eigenvalues and eigenvectors

Many people have vague memories of their discrete mathematics classes. Everybody seems to remember something about eigenvalues, eigenvectors, and that it had something to do with $\lambda$? ('lambda') in order to compute them. If you want to fresh up about eigenvalues and eigenvectors, then this page may be something for you. I present what eigenvalues and eigenvectors are and how you can compute them.

Eigenvectors and eigenvalues

Definition

Consider a square $n\times n$? matrix $A$?. A column vector text/plain$? is an eigenvector of matrix $A$? if $Av=\lambda v$? with $\lambda$? a scalar. This lambda is the eigenvalue associated with eigenvector text/plain$?.

Importance

Eigenvectors are pretty interesting vectors that can help you studying properties of the matrix. As an example, imagine that the matrix denotes a coordinate transformation. The eigenvectors than denote the points of your space that are simply scaled by the transformation, by definition. If you can identify eigenvectors that have a corresponding eigenvalue 1, than the vector is unchanged by the transfromation, also by definition. In Finite Markov Models, the eigenvectrs with assciated eigenvalue 1 are important as they denote the fixed points (or limiting) of the system, which is actually also a transformation.

Computing eigenvalues and eigenvectors

Computing eigenvalues with the characteristic function

An eigenvector text/plain$? and its eigenvalue $\lambda$? of matrix $A$? were defined with the property that $Av-\lambda v=0$?. We can easily find the eigenvalues by solving A-\lambda I|=0$, where \cdot|$ denotes the determinant of the dot. $A-\lambda I$? is the characteristic function of our matrix, and by determining its roots, we can find the eigenvalues.

I assume you know how to compute the determinant of a matrix.

Example

Consider matrix $A=\left$?. In order to find its eigenvalues, we have to solve \left(\begin{array}{c}0\hspace{4mm}1\\-2\hspace{4mm}-3\end{array}\right)-\lambda\left(\begin{array}{c}1\hspace{4mm}0\\0\hspace{4mm}1\end{array}\right)\right|=0\?

\begin{array}{c}0-\lambda\hspace{14mm}1\\-2\hspace{14mm}-3-\lambda\end{array}\right|=0\?

\\lambda+2=\lambda^2+3\lambda+2=0\?

The roots of our equation are $\lambda_1=-1$? and $\lambda_2=-2$?. These now denote the eigenvalues of our matrix.

Computing eigenvectors