title: Eigenvalues and eigenvectors
taxon: Definition
parent: linear-algebra
math: yes

A nonzero vector $v$ is an eigenvector of a square matrix $A$ with
eigenvalue $\lambda$ when

$$
A v = \lambda v .
$$

The eigenvalues are the roots of the characteristic polynomial
$\det(A - \lambda I) = 0$. A matrix with $n$ independent eigenvectors
can be diagonalised, $A = P D P^{-1}$, which is what makes powers of
$A$, and so things like Markov chains in [[operating-systems|scheduling]]
models, easy to compute.