Relation between the matrix trace and the amplitude of each element
The trace of a matrix just scales with the scaling of the matrix:
$$\textrm{tr}\{c\mathbf{A}\}=c\,\textrm{tr}\{\mathbf{A}\}\tag{1}$$
So if you scale the vector $\mathbf{p}$, the trace scales with the squared magnitude of the scaling constant. With $\mathbf{p}=c\mathbf{p}_0$ you have
$$\begin{align}\textrm{tr}\{\mathbf{Ap}\mathbf{p}^H\mathbf{A}^H\}&amp;=\textrm{tr}\{\mathbf{A}c\mathbf{p}_0c^*\mathbf{p}_0^H\mathbf{A}^H\}\\&amp;=\textrm{tr}\{|c|^2\mathbf{A}\mathbf{p}_0\mathbf{p}_0^H\mathbf{A}^H\}\\&amp;=|c|^2\textrm{tr}\{\mathbf{A}\mathbf{p}_0\mathbf{p}_0^H\mathbf{A}^H\}\end{align}$$