Let $W = aI\_{n\times n} + bJ\_{n\times n}$, where $I$ is an identity matix, $J$ is the matrix of all ones, $a,b\in\mathbb{R}$ and a+b>0. Also, let $A = \mathbf{P} - \mathbf{p}\mathbf{p}^{T}$, where $\mathbf{p} = (p\_{1},\ldots,p\_{n})^{T}$ and $\mathbf{P} = \rm{diag}(\mathbf{p})$ with $\sum\_{i=1}^{n}p\_i = 1$ and $0<p\_i<1$. Is it possible to show that $trace((AW)^K)$ for any $K>1$ is symmteric function of $\{p\_1,\ldots,p\_k\}$?


I tried for $K=2,3$ and its true but in general not able to show.



Proving symmetry of trace function of special matrix
To check the symmetry of the trace under a given permutation of the $p\_i$'s, take the corresponding permutation matrix $S$ and insert in the trace noting that $S^TS=I$ and $SWS^T=W$:


$${\rm tr}\,(AW)^K={\rm tr}\,(AS^T SWS^TS)^K={\rm tr}\,(\tilde{A}W)^K$$


with $\tilde{A}=SAS^T$ the matrix $A$ after the desired permutation of the $p\_i$'s.


