  We study an eigenvalue problem for the Laplacian on a compact K\"{a}hler
manifold. Considering the $k$-th eigenvalue $\lambda_{k}$ as a functional on
the space of K\"{a}hler metrics with fixed volume on a compact complex
manifold, we introduce the notion of $\lambda_{k}$-extremal K\"{a}hler metric.
We deduce a condition for a K\"{a}hler metric to be $\lambda_{k}$-extremal. As
examples, we consider product K\"{a}hler manifolds, compact isotropy
irreducible homogeneous K\"{a}hler manifolds and flat complex tori.
