  Let $(E,\Phi)\rightarrow (X,\omega_X)$ be a Higgs bundle over a compact
K\"ahler manifold. We suppose that the holomorphic vector bundle $E$ decomposes
into a direct sum of holomorphic line bundles. In this paper, we give the
necessary and sufficient condition for the existence of a diagonal metric which
is a solution to the Hermitian-Einstein equation. Our theorem can easily be
generalized to $G$-Higgs bundles. We also describe the relationship between the
stability condition and our condition using the torus action on the space of
Higgs fields.
