@MISC{Qi07eigenvaluesand, author = {Liqun Qi}, title = { Eigenvalues and invariants of tensors}, year = {2007} }

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Abstract

A tensor is represented by a supermatrix under a coordinate 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 invari-ants 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) mn−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.