Some basic observations lead me to ask the following quesiton


Let $A\_1, \cdots, A\_m$ be $n\times n$ complex matrices. For positive integer $k\ge 1$, show
$$\left(\begin{array}{cccc}Tr\{(A\_1^\*A\_1)^k\}&Tr\{(A\_1^\*A\_2)^k\}&\cdots &Tr\{(A\_1^\*A\_m)^k\}\\Tr\{(A\_2^\*A\_1)^k\}&Tr\{(A\_2^\*A\_2)^k\}&\cdots &Tr\{(A\_2^\*A\_m)^k\}\\\cdots&\cdots&\cdots&\cdots\\Tr\{(A\_m^\*A\_1)^k\}&Tr\{(A\_m^\*A\_2)^k\}&\cdots &Tr\{(A\_m^\*A\_m)^k\}
\end{array}\right)$$
is positive semidefinite.


**Remark**


1). When $m=2$, it suffices to show $|Tr\{(A\_1^\*A\_2)^k\}|^2\le Tr\{(A\_1^\*A\_1)^k\}\cdot Tr\{(A\_2^\*A\_2)^k\}$, which is a consequence of a unitarily invariant norm inequality appeared in p.81 of X.Zhan, Matrix inequalities, Springer, 2002.


2). It is easy to show $$\left(\begin{array}{cccc}(Tr\{A\_1^\*A\_1\})^k&(Tr\{A\_1^\*A\_2\})^k&\cdots &(Tr\{A\_1^\*A\_m\})^k\\(Tr\{A\_2^\*A\_1\})^k&(Tr\{A\_2^\*A\_2\})^k&\cdots &(Tr\{A\_2^\*A\_m\})^k\\\cdots&\cdots&\cdots&\cdots\\(Tr\{A\_m^\*A\_1\})^k&(Tr\{A\_m^\*A\_2\})^k&\cdots &(Tr\{A\_m^\*A\_m\})^k
\end{array}\right)$$
is positive semidefinite, since it is $k$ Hadamard product of a Gram matrix.



On a positivity of a matrix with trace entries.
It seems that this is not true. Here is a counterexample. Consider a regular $d$-gon, where $d\ge 3$ is an odd number. Let $S,T$ be the permutation matrices on the vertices of this $d$-gon, induced by reflections in two adjacent symmetry axes. Let $A\_1=1, A\_2=S, A\_3=T$, which are $d$ by $d$ matrices. We have $S^2=T^2=1$, and $ST$ is a rotation of order $d$
(so $(ST)^2$ has no fixed points). 


Let $k=2$. Then the matrix in the question seems to be the following:
$$
\left(\begin{matrix} d & d & d\\ d & d & 0\\ d & 0 & d\\ \end{matrix}\right)
$$
The determinant of this matrix is $-d^3$. 


