I am looking for a general formula for the trace of monomials of the complex matrices $X=A+A^T$ and $P=i(-A+A^T)$, where $$ A=\begin{pmatrix} 0 &\sqrt{1} & 0 & 0 & \dots & 0 \\ 0 & 0 & \sqrt{2} & 0 & \dots & 0 \\ 0 & 0 & 0 &\sqrt{3} & \dots & 0 \\ 0 & 0 & 0 & 0 & \ddots &\vdots \\ \vdots & \vdots & \vdots &\vdots & \ddots &\sqrt{n-1} \\ 0 & 0 &0 &0 &\dots &0 \end{pmatrix}. $$
I became intrigued by the algebraic patterns exhibited by traces of monomials in $X$ and $P$. Before list them, let me define the degree in $X$ (or $P$) of a matrix $M$ written as monomial, $\text{deg}_X(M)$, as the the sum of exponents of $X$. For example, $\text{deg}_X(X^2PX^3P^7X)=6$ and $\text{deg}_P(X^2PX^3P^7X)=8$. Now a partial list:
- The trace is nonzero only if the degree in $X$ and $P$ is even and it is always an integer.
- The trace is invariant if $X$ and $P$ are exchanged. So, for example, $\text{trace}(P^2)=\text{trace}(X^2)$.
- For $3\times 3$ matrices, $\text{trace}(X^{2n})=\text{trace}(P^{2n})=2\cdot 3^n$, where $n$ is positive. There is no direct generalization for bigger matrices. For $4\times 4$ matrices, for example, $\text{trace}(X^{2n})= 12, 60, 324, 1764$ with $n=1,2,3,4$.
- For a $n\times n$ matrix monomial, $\text{trace}(X^2)=n\cdot(n-1)$.
- For $2\times 2$ matrix monomials, the nonzero traces are always $\pm 2$.
If the degree in $X$ and $P$ are fixed, the monomials have different nonzero traces only if they cannot be transformed into each other by a cyclic permutation or an exchange of $X$ and $P$. Thus, for example $\text{trace}(XPXP)=-2$ and $\text{trace}(X^2P^2)=10$ for $3\times 3$ matrix monomials.