General Results on Positive and Negative Definite Matrices and Kernels

Abstract

When dealing with positive and negative definite kernels a certain amount of confusion often arises concerning terminology. A positive definite kernel defined on a finite set is usually called a positive semidefinite matrix. Sometimes it is only called “positive”, which may be misleading. When working on groups, the name positive definite function is used traditionally. In our previous papers on abelian semigroups we also followed this tradition. Instead of calling a kernel ψ negative definite, some authors call the kernel —ψ “conditionally positive definite” or “almost positive.” In this book we use mainly the larger class of “semidefinite” kernels of all kinds and therefore prefer to avoid the prefix “semi” which otherwise would appear several hundred times.