I need to compute the determinant of a matrix that is calculated as $B \circ A$, with $B$ and $A$ being square matrices and $\circ$ representing their Hadamard product.
One way of doing this is through the eigenvalues of $B \circ A$, since a determinant is equal to the product of eigenvalues. This calculation is part of a simulation, where $A$ remains constant and $B$ changes during the simulation. Since $A$ does not change, its eigenvalues will not change, but $B$'s eigenvalues will.
Is there a way to calculate this determinant by only updating the eigenvalues of $B$? Or, similarly, how can I efficiently determine the eigenvalues of the Hadamard product of two matrices (perferably in R)?