Let $V$ be an $n$-dimensional [vector space](/page/Vector%20Space) over a field $k$, let $T:V\to V$ be a [linear map](/page/Linear%20Map), and let $\mathcal B=(v_1,\ldots,v_n)$ be an ordered basis of $V$. Then the matrix representation $[T]_{\mathcal B \leftarrow \mathcal B}$ is diagonal if and only if $\mathcal B$ is an eigenbasis for $T$, meaning that each basis vector $v_i$ is an eigenvector of $T$.