The E-characteristic polynomial of a tensor of dimension 2

We show that the E-characteristic polynomial ψT(λ)ψT(λ) of a tensor TT of order m≥3m≥3 and dimension 2 is ψT(λ)=det(S−λT)ψT(λ)=det(S−λT) with SS a variant of the Sylvester matrix of the system Txm−1=0, and TT a constant matrix that is only dependent on mm. By exploring special structures of the matrices SS and TT, the coefficients of the E-characteristic polynomial ψT(λ)ψT(λ) which make the computation of ψT(λ)ψT(λ) efficient are obtained. On the basis of these, we prove that the leading coefficient of ψT(λ)ψT(λ) is (pm2+qm2)m−22 when mm is even and −(pm2+qm2)m−2 when mm is odd, which strengthens Li, Qi and Zhang’s theorem.