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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 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 Phys.SE post.

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

$$\begin{align}S~=~& {\rm Tr} L(H), \cr L(H)~=~&\sum_{n\in\mathbb{N}_0}c_n H^n,\cr c_n~\in~&\mathbb{R},\end{align}\tag{1}$$

is invariant under adjoint conjugation

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

with unitary matrices $U$. Eq.(2) here plays 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

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

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 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 Phys.SE post.

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 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 Phys.SE post.

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 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 Phys.SE post.

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.

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 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 Phys.SE post.