The E-characteristic polynomial of a tensor of dimension 2

We show that the E-characteristic polynomial ψτ(λ ) of a tensor T of order m<3 and dimension 2 is ψτ(λ)=det(S-λT) with S a variant of the Sylvester matrix of the system Txm-1=0, and T a constant matrix that is only dependent on m. By exploring special structures of the matrices S and T, the coefficients of the E-characteristic polynomial ψτ(λ) which make the computation of ψτ(λ) efficient are obtained. On the basis of these, we prove that the leading coefficient of ψτ(λ) is (pm2+qm2)m-2/2 when m is even and -( pm2+qm2)m-2when m is odd, which strengthens Li, Qi and Zhang's theorem.