P-tensors, P₀-tensors, and their applications

P- and P₀-matrix classes have wide applications in mathematical analysis, linear and nonlinear complementarity problems, etc., since they contain many important special matrices, such as positive (semi-)definite matrices, M-matrices, diagonally dominant matrices, etc. By modifying the existing definitions of P- and P₀-tensors that work only for even order tensors, in this paper, we propose a homogeneous formula for the definition of P- and P₀-tensors. The proposed P- and P₀-tensor classes coincide the existing ones of even orders and include many important structured tensors of odd orders. We show that many checkable classes of structured tensors, such as the nonsingular M-tensors, the nonsingular H-tensors with positive diagonal entries, the strictly diagonally dominant tensors with positive diagonal entries, are P-tensors under the new definition, regardless of whether the order is even or odd. In the odd order case, our definition of P₀-tensors, to some extent, can be regarded as an extension of positive semi-definite (PSD) tensors. The theoretical applications of P- and P₀-tensors under the new definition to tensor complementarity problems and spectral hypergraph theory are also studied.