Abstract :

An mm-order nn-dimensional square real tensor AA is a multidimensional array of nmnm elements of the form {A=(Ai1…im)A=(Ai1…im), Ai1…im RAi1…im R, 1≤i1, ∈ ∈ …,im≤n.1≤i1,…,im≤n.} (A square matrix of order nn is a 22-order nn-dimensional square tensor). An mm-order nn-dimensional square real tensor is said to be a nonnegative (positive) tensor if all its entries are nonnegative (positive). We shall discuss the Perron-Frobenius theory for nonnegative tensors. Using these results we establish a sufficient condition for the positive semidefiniteness of homogenous multivariable polynomials.