On a positivity of a matrix with trace entries.

Question

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.

Answer 1

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 $-d^3$.