The E-characteristic polynomial of a tensor of dimension 2

We show that the E-characteristic polynomial ψ T ( λ ) of a tensor T of order m ≥ 3 and dimension 2 is ψ T ( λ ) = det ( S − λ T ) with S a variant of the Sylvester matrix of the system T x m − 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 ψ T ( λ ) which make the computation of ψ T ( λ ) efficient are obtained. On the basis of these, we prove that the leading coefficient of ψ T ( λ ) is ( p m 2 + q m 2 ) m − 2 2 when m is even and − ( p m 2 + q m 2 ) m − 2 when m is odd, which strengthens Li, Qi and Zhang’s theorem.