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Eigenvalues and eigenvectors

An eigenvector $\vec v$ of the matrix $A$ is a vector that is only scaled, not rotated, when multiplied by $A$ . The scaling factor $\lambda$ is called the eigenvalue.

A\vec v = \lambda\vec v,\quad \vec v\neq\vec 0

the eigenvalue equation

\det(A-\lambda I) = 0

characteristic equation, gives the eigenvalues

Symbols

$\lambda$	eigenvalue
$\vec v$	eigenvector

Example

$A=\begin{pmatrix}4&1\\2&3\end{pmatrix}$ : $\lambda^2-7\lambda+10=0$ gives $\lambda=5$ and $\lambda=2$ .

The sum of the eigenvalues is the trace of

A

, and the product is

\det A

- a quick check.

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Part of Linear Algebra and Differential Equations: Linear systems and eigenvalues.