  Let $\mathfrak{g}$ be a finite-dimensional complex Lie algebra and
$\textrm{HLie}_{m}(\mathfrak{g})$ be the affine variety of all multiplicative
Hom-Lie algebras on $\mathfrak{g}$. We use a method of computational ideal
theory to describe $\textrm{HLie}_{m}(\mathfrak{gl}_{n}(\mathbb{C}))$, showing
that $\textrm{HLie}_{m}(\mathfrak{gl}_{2}(\mathbb{C}))$ consists of two
1-dimensional and one 3-dimensional irreducible components, and showing that
$\textrm{HLie}_{m}(\mathfrak{gl}_{n}(\mathbb{C}))=\{\textrm{diag}\{\delta,\dots,\delta,a\}\mid
\delta=1\textrm{ or }0,a\in\mathbb{C}\}$ for $n\geqslant 3$. We construct a new
family of multiplicative Hom-Lie algebras on the Heisenberg Lie algebra
$\mathfrak{h}_{2n+1}(\mathbb{C})$ and characterize the affine varieties
$\textrm{HLie}_{m}(\mathfrak{u}_{2}(\mathbb{C}))$ and
$\textrm{HLie}_{m}(\mathfrak{u}_{3}(\mathbb{C}))$. We also study the derivation
algebra $\textrm{Der}_{D}(\mathfrak{g})$ of a multiplicative Hom-Lie algebra
$D$ on $\mathfrak{g}$ and under some hypotheses on $D$, we prove that the
Hilbert series $\mathcal{H}(\textrm{Der}_{D}(\mathfrak{g}),t)$ is a rational
function.
