Positive definite functions arise in various areas of mathematical analysis such as classical interpolation problems, the Loewner theory of monotone operator functions, operator inequalities and others. The present paper has two objectives. The first is to present and to advance a technique for proving the positive definiteness of some classes of matrices related to important problems in operator theory. The second is to show how this positivity can be used to prove inequalities for norms of operators. The following ideas underly the method suggested by the authors. Starting from a matrix \(X\) with entries of the form \(x_{ij}=f(\lambda_i, \lambda_j)\) they find a congruence that converts it to \(Y\) with entries \(y_{ij}=g(\lambda_i-\lambda_j)\). Next, the fundamental Bochner theorem shows whether the function \(g\) is positive definite. Finally, \(X\) is used as a Schur multiplier to prove certain operator inequalities.