Showing that a certain complex matrix is positive definite
The common denominator does not affect the result, so it suffices to show that the matrix $A = (a_{ij})$ is positive definite. $A$ can be written as a rank-one modification of a diagonal matrix: $$ A = (1 + |z|^2) I - \overline z z^T $$ where $I$ is the $n$-dimensional identity matrix and $\overline z$ denotes the component-wise complex conjugate of the column vector $z$. Then $$ w^* A w = (1 + |z|^2) |w|^2 - \overline w^T z z^T w = (1 + |z|^2) |w|^2 - |z^T w|^2 \ge |w|^2 > 0 $$ for all non-zero vectors $w$, using the Cauchy-Schwarz inequality.