Two classes of element-wise transformations preserving the psd nature of coefficient matrices

Two classes of element-wise transformations are proved to preserve the positive semi-definite nature of coefficient matrices. The correctness of a conjecture by Gower and Legendre on the positive semidefinite nature of a certain coefficient matrix is proved. It is shown that the matrix of monotonicity coefficients proposed by Bentler is positive semidefinite for data without ties.