Eigensystem

The eigensystem of a square matrix \(\mathbf{M}\) is the set of vectors \(\mathbf{v}_i\), known as eigenvectors, and numbers \(\lambda_i\), known as eigenvalues, that satisfy the relation
\[ \mathbf{M} \times \mathbf{v}_i = \lambda_i \times \mathbf{v}_i \]
The eigenvectors are thus those vectors whose direction is unchanged when multiplied by the matrix.
We typically restrict the set of vectors to those of unit length since multiplying an eigenvector by a scalar results in another eigenvector with the same direction
\[ \begin{align*} \mathbf{M} \times c \times \mathbf{v}_i &= c \times \mathbf{M} \times \mathbf{v}_i\\ &= c \times \lambda_i \times \mathbf{v}_i\\ &= \lambda_i \times c \times \mathbf{v}_i \end{align*} \]