lp/2,q/2-Singular values of a real partially symmetric rectangular tensor

Let A be a real ( p , q )-th order m × n dimensional partially symmetric rectangular tensor with p and q even. Firstly, in order to judge the positive definiteness of A , an l p / 2 , q / 2 -singular value inclusion interval with parameters is constructed. Subsequently, by finding the optimal values of parameters, the optimal parameter inclusion interval of l p / 2 , q / 2 -singular values is derived, which provides a sufficient condition for the positive definiteness of A . Secondly, when A is a nonnegative tensor, lower and upper bounds for its l p / 2 , q / 2 -spectral radius are given. Thirdly, a direct method for finding all l p / 2 , q / 2 -singular value/vectors pairs of A with p = q = 4 and m = n = 2 is presented. Finally, a numerical example is given to verify the theoretical results.