I have a matrix A with null trace. What is the minimum number of linear measurements that I hav

I have a matrix $A$ with null trace.
What is the minimum number of linear measurements that I have to perform in order to determine $A$?
By a linear measurement I mean that I know the quantities $A{\stackrel{\to }{x}}_{i}\cdot {\stackrel{\to }{h}}_{j}$ for given ${\stackrel{\to }{x}}_{i}$ and ${\stackrel{\to }{h}}_{j}$
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Using unit vectors for both parts of the measurement you have ${e}_{i}^{T}A{e}_{j}={A}_{ij}$ so if “one measurement” corresponds to one unique pair $\left({\stackrel{\to }{x}}_{i},{\stackrel{\to }{h}}_{j}\right)$ then each measurement of this form will yield one entry of the matrix. For a generic $m×n$ matrix you'd therefore need $m\cdot n$ measurements. Knowing the trace adds one linear constraint, so you can infer one value from all the others, i.e. $m\cdot n-1$ measurements. I can not imagine how a different choice of measurement vectors could do better than this.
On the other hand, where I wrote $\left({\stackrel{\to }{x}}_{i},{\stackrel{\to }{h}}_{j}\right)$ you had $\left({\stackrel{\to }{x}}_{i},{\stackrel{\to }{h}}_{j}\right)$. Were you thinking of combining a single $x$ vector with multiple $h$ vectors, or vice versa? If so, you will need $m$ vectors ${\stackrel{\to }{h}}_{j}$ and $n$ vectors ${\stackrel{\to }{x}}_{i}$, minus one combination due to the trace. I'll leave it to you to decide what you count as “one measurement” in this case.