Eigenvalues and invariants of tensors

Abstract

A tensor is represented by a supermatrix under a co-ordinate system. In this paper, we define E-eigenvalues and E-eigenvectors for tensors and supermatrices. By the resultant theory, we define the E-characteristic polynomial of a tensor. An E-eigenvalue of a tensor is a root of the E-characteristic polynomial. In the regular case, a complex number is an E-eigenvalue if and only if it is a root of the E-characteristic polynomial. We convert the E-characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor, i.e., they are invariant under co-ordinate system changes. We call them principal invariants of that tensor. The maximum number of principal invariants of mth order n-dimensional tensors is a function of m and n. We denote it by d ( m , n ) and show that d ( 1 , n ) = 1 , d ( 2 , n ) = n , d ( m , 2 ) = m for m ⩾ 3 and d ( m , n ) ⩽ m n − 1 + ⋯ + m for m , n ⩾ 3 . We also define the rank of a tensor. All real eigenvectors associated with nonzero E-eigenvalues are in a subspace with dimension equal to its rank.