Suppose that $\ltdefn{T}{V}{V}$ is a linear transformation and $B$ is a basis of $V$. Then $\vect{v}\in V$ is an eigenvector of $T$ for the eigenvalue $\lambda$ if and only if $\vectrep{B}{\vect{v}}$ is an eigenvector of $\matrixrep{T}{B}{B}$ for the eigenvalue $\lambda$.