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 scheduling models, easy to compute.