Let $A,B,C$ complex $n\times n$ Matrices with ones on the diagonal and entries of absolute value 1. Further, let $$A=(a_{ij})_{i,j=1}^n,\; B=(b_{ij})_{i,j=1}^n \mbox{ and } C=(a_{ij}b_{ij})_{i,j=1}^n.$$
Can we say something about the positive semi-definiteness of $A$ if we know that $B$ and $C$ have this property?
Any hint to a similar problem or result or counter-example would be greatly appreciated.
If by complex matrices you mean Hermitian matrices the answer is yes, for a silly reason and a non-silly reason. Suppose $B$ and $C$ are PSD with $1$s on the diagonal and all off diagonal entries of absolute value $1$. You ask if the elementwise quotient $A=(c_{i,j}/b_{i,j})$ is PSD. The silly reason is that the complex conjugate of a complex number of absolute value $1$ is its reciprocal. So the matrix $D=(\bar{b}_{i,j})$ is the matrix of reciprocals of the entries in $B$, and $A$ is the element-wise product of the two PSD matrices $A$ and $D$. Which it is also PSD, by the non-silly Schur product theorem: https://en.wikipedia.org/wiki/Schur_product_theorem.