Most hyper-ellipsoidal clustering(HEC) algorithms use the Mahalanobis distance as a distance metric. It has been proven that HEC, under this condition, cannot be realized since the cost function of partitional clustering is a constant. We demonstrate that HEC with a modified Gaussian kernel metric can be interpreted as a problem of finding condensed ellipsoidal clusters(with respect to the volumes and densities of the clusters) and propose a practical HEC algorithm named K-HEC that is able to efficiently handle clusters that are ellipsoidal in shape and that are of different size and density. We then try to refine the K-HEC algorithm by utilizing ellipsoids defined on the kernel feature space to deal with more complex-shaped clusters. Simulation experiments demonstrate the proposed methods have a significant improvement in the clustering results and performance over K-means algorithm, fuzzy C-means algorithm, GMM-EM algorithm and HEC algorithm based on minimum-volume ellipsoids using Mahalanobis distance.