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The concepts of P- and P$_0$-matrices are generalized to P- and P$_0$-tensorsof even and odd orders via homogeneous formulae. Analog to the matrix case, ourP-tensor definition encompasses many important classes of tensors such as thepositive definite tensors, the nonsingular M-tensors, the nonsingular H-tensorswith positive diagonal entries, the strictly diagonally dominant tensors withpositive diagonal entries, etc. As even-order symmetric PSD tensors are exactlyeven-order symmetric P$_0$-tensors, our definition of P$_0$-tensors, to someextent, can be regarded as an extension of PSD tensors for the odd-order case.Along with the basic properties of P- and P$_0$-tensors, the relationship amongP$_0$-tensors and other extensions of PSD tensors are then discussed forcomparison. Many structured tensors are also shown to be P- and P$_0$-tensors.As a theoretical application, the P-tensor complementarity problem is discussedand shown to possess a nonempty and compact solution set.

P-Tensors, P$_0$-Tensors, and Tensor Complementarity Problem