Eigenvector with Eigenvalue

Let $V$ be a \reftext{def:vector-space-2026a}{complex vector space} with \reftext{lem:vector-space-basic-identities-2026a}{zero vector} $0_{V}$, let $T$ be a \reftext{def:linear-operator-2026a}{linear operator} on $V$, let $\lambda$ be a \reftext{def:complex-numbers-2026a}{complex number}, and let $x\in V$.

The vector $x$ is an \textbf{eigenvector of $T$ with eigenvalue $\lambda$} if $x\ne 0_{V}$ and