A tensor singular value decomposition based upon the *-product defined on third-order tensors by Kilmer,Martin and Perrone provides an effective means of generalizing Principal Component Analysis.This talk will focus on tensor eigendecompostions and generalized eigendecompostions under the *-product.Discusion will include an overview of the theoretical underpinings,some computational concerns and also the use of tensor eigendecompositions in a generalization of Fischers Linear Discriminant Analysis as applied to object recognition.