Validity of Clusters Produced By kernel-k-means With Kernel-Trick

This paper corrects the proof of the Theorem 2 from the Gower's paper cite[page 5]Gower:1982 as well as corrects the Theorem 7 from Gower's paper citeGower:1986. The first correction is needed in order to establish the existence of the kernel function used commonly in the kernel trick e.g. for k-means clustering algorithm, on the grounds of distance matrix. The correction encompasses the missing if-part proof and dropping unnecessary conditions. The second correction deals with transformation of the kernel matrix into a one embeddable in Euclidean space.