In this answer we will basically repeat Lubos Motl's answer in other words and introduce some terminology in the process.
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 $U=e^{\epsilon}$, the gauge transformation is
$$\tag{3} \delta H ~=~[\epsilon, H],$$
where $\epsilon$ 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 $\epsilon^1{}_1$ , $\epsilon^2{}_2$, $\ldots$, $\epsilon^N{}_N$, at the infinitesimal level.