Let S^pV be the p-th symmetric power of a complex vector space V.A vector v in V is called a (generalized) eigenvector of a linear map M: S^pV --> V,if M(v^p)=aV for some scalar a,called eigenvalue.When a=1 this notion coincides with the usual definition of eigenvector.We show how eigenvectors are useful in the problem of tensor decomposition (how to decompose a tensor in the sum of decomposable ones) and we sketch the connection with the complexity of the matrix multiplication algorithm.