Eigenvectors and Eigenvalues

An eigenvector of a square matrix $A$ is a vector ${\boldsymbol {x}}\in \mathbb {R}$ such that,

$A{\boldsymbol {x}}=\lambda {\boldsymbol {x}}$

where $\lambda$ is complex number, known as a eigenvalue. The eigenvalue is the zero of a matrix's charateristic polynomial.

All the eigenvectors of a particular eigenvalue span a vector space called the eigenspace.