Positive definiteness of infinite and finite dimensional generalized Hilbert tensors and generalized Cauchy tensor

An Infinite and finite dimensional generalized Hilbert tensor with a is positive definite if and only if a>0. The infinite dimensional generalized Hilbert tensor related operators F∞ and T∞ are bounded, continuous and positively homogeneous. A generalized Cauchy tensor of which generating vectors are c,d is positive definite if and only if every element of vector d is not zero and each element of vector c is positive and mutually distinct. The 4th order n-dimensional generalized Cauchy tensor is matrix positive semi-definite if and only if every element of generating vector c is positive. Finally, the other properties of generalized Cauchy tensor are presented.