Let $H$ be a "regular" $n$ by $n$ hermitian matrix, where by "regular", we mean that it has distinct eigenvalues. We can then write $H$ as $$ H = UDU^*, $$ where $U$ is unitary and $D$ is a real diagonal matrix containing the eigenvalues of $H$ on its diagonal. WLOG, assume that the elements on the diagonal of $D$ are ordered by decreasing order ($D_{11} > D_{22} > \ldots > D_{nn}$).
We are interested in the following operation. Given $\sigma \in S_n$, where $S_n$ is the symmetric group on $n$ elements, we denote by $P_{\sigma}$ the corresponding permutation matrix. Actually, we will drop the $\sigma$ from the subscript, and just denote by $P$ the corresponding permutation matrix.
Define $$ H_{\sigma} := U P^{-1} D P U^*. $$
The "operation" we are interested in is the following. Given $\sigma \in S_n$, we map each regular hermitian matrix $H$ to $H_{\sigma}$. Note that each $H_{\sigma}$ is also a regular hermitian matrix.
We can either think of $H_{\sigma}$ as the regular hermitian matrix having the same fixed eigenvalues as $H$ but with the eigenvectors having been "actively" permuted via $\sigma$, or equivalently, as the regular hermitian matrix having the same fixed set of eigenvectors as $H$, but with the eigenvalues having been "passively" permuted via $\sigma$ (I am using the active/passive terminology borrowed from physicists, well, at least the way I understand it).
Mathematically, you could think that $D$ is fixed and that $U \mapsto U P^{-1}$ or, equivalently you could think that $U$ is fixed and that $D \mapsto P^{-1} D P$.
What is known about such operations? Are there some known formulas for them perhaps, that do not involve first doing an eigendecomposition of $H$? Could someone perhaps provide some references?
I came across such operations towards the end of an article by Berry and Robbins entitled "Indistinguishability for quantum particles: spin, statistics and the geometric phase". I have a feeling they have been studied before though.
Edit: if one removes the requirement that $H$ be hermitian and assumes only that $H$ is diagonalizable and has distinct eigenvalues (which are in general complex), similar questions may be asked.
let $A$ (changing the notation from $H$ to $A$) be a diagonalizable complex square matrix with distinct eigenvalues. One may write $$A = GDG^{-1},$$ for some nonsingular $G$ and a diagonal matrix $D$. The entries of $D$ in its diagonal are then the eigenvalues of $A$.
There is then no natural way to order the eigenvalues in general. But we may still label them in some way as $\lambda_1, \ldots, \lambda_n$ and assume WLOG that $D = \operatorname{diag}(\lambda, \ldots, \lambda_n)$. Then, given $\sigma \in S_n$, one may define $$ A_{\sigma} = GP^{-1}DPG^{-1}.$$
In general, I have a feeling that there is no direct way to describe how to go from $A$ to $A_{\sigma}$, given $\sigma \in S_n$. However, let us assume we are in the special case where the eigenvalues of $A$ are the $n$-th roots of unity. Then, in this case, if $\sigma = (1 2 \cdots n)$ and unless I made a silly mistake, $$ A_{\sigma} = e^{-\frac{2 \pi i}{n}} A.$$
I'd also be happy with a description of $A_{\sigma}$ for any $\sigma \in S_n$ in the special case where the eigenvalues of $A$ are the $n$-th roots of unity.