Definition
Eigenvalues can be interpreted as the scaling coefficients of the related eigenvectors.
They are defined by the solution of n-th degree polynomials .
Also known as characteristic roots and latent roots.
Complex eigenvalues
Sometimes this polynomial have complex roots. In such case one should only find one root and calculate other root as . Therefore we can say that complex eigenvalues and eigenvectors appear in complex conjugate pairs.
\begin{align*} Av &= \lambda v \ \overline{Av} &= \overline{\lambda v} \ A \bar{v} &= \bar{\lambda} \bar{v} \end{align*}