  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 LSDs
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 semi-circular LSD. We also prove an invariance
theorem that may be used to find the LSD in many examples.
