Eigenvectors and Eigenvalues

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Eigenvectors represent directions that are preserved by linear transformations of a vector space.

An eigenvector of an n x n square matrix $A$ is a vector ${\boldsymbol {x}}\in \mathbb {C} ^{n}$ 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.