A Generalized Spectral Theory for Tensors Technical Report DCS-tr-663

In this paper we proposes a generalized spectral theory for tensors. Our proposed factorization decomposes a symmetric tensor into a product of an orthogonal and a diagonal tensor. In the same time, our factorization offers an expansion of a tensor as a summation of lower rank tensors that are obtained through an outer product defined on matrices. Our proposed factorization shows the relationship between the eigen-objects (eigen matrices and eigen vectors for order 3 tensor) and the generalized determinant and trace operators. Our framework is based on a consistent multilinear algebra that explains how we can generalize the notion of Hermitian matrices, the notion of transpose, and most importantly the notion of orthogonality for tensors.