In this answer we will basically expand on Lubos Motl's correct answer using some other words and introducing some terminology. 

In the Hermitian one-matrix model, the action 

$$\tag{1}S~=~ {\rm Tr} L(H)$$ 

is invariant under adjoint conjugation 

$$\tag{2} H\to UHU^{-1}$$ 

with unitary matrices $U$. Eq.(2) here play the role of the gauge transformations.  One therefore has $N^2$ real gauge parameters. 

At the infinitesimal level $U=e^A$, the gauge transformation is 

$$\tag{3} \delta H ~=~[A, H],$$ 

where $A$ is an infinitesimal anti-Hermitian matrix.

On the other hand, the $N$ real eigenvalues $\lambda_1$, $\ldots$, $\lambda_N$ of an Hermitian matrix are gauge invariants, which cannot be changes by gauge transformations of adjoint type (2). Hence there are actually only $N(N−1)$ independent gauge parameters. Such a gauge algebra is called _reducible._

For a diagonal Hermitian matrix $H$, the $N$ redundant gauge parameters may be identified with the diagonal matrix entries $A^1{}_1$ , $A^2{}_2$, $\ldots$, $A^N{}_N$, at the infinitesimal level.

The Faddeev-Popov trick (in its original formulation) applies to _irreducible_ gauge symmetries, but one can make it work in this reducible case (i.e. the Hermitian one-matrix model) by properly identifying the independent gauge parameters, cf. above. 

The Faddeev-Popov determinant becomes the square of the [Vandermonde determinant](http://en.wikipedia.org/wiki/Vandermonde_matrix) of the eigenvalues $\lambda_1$, $\ldots$, $\lambda_N$. 

Finally, it seems natural to mention that the presence of this Vandermonde determinant is a typical feature of (random) matrix models, and it leads to eigenvalue repulsion, cf. e.g. [this](https://physics.stackexchange.com/q/32041/2451) Phys.SE post.