Bulk behavior of Schur-Hadamard products of symmetric random matrices

We develop a general method for establishing the existence of the Limiting Spectral Distributions (LSD) of Schur-Hadamard products of independent symmetric patterned random matrices. We apply this method to show that the LSD of Schur-Hadamard products of some common patterned matrices exist and identify the limits. In particular, the Schur-Hadamard product of independent Toeplitz and Hankel matrices has the semicircular LSD. We also prove an invariance theorem that may be used to find the LSD in many examples.