A direct proof of uniqueness of square-root of a positive semi-definite tensor

Understanding of the basic properties of the positive semi-definite tensor is a prerequisite for its extensive applications in theoretical and practical fields,especially for its square-root.Uniqueness of the square-root of a positive semi-definite tensor is proven in this paper without resorting to the notion of eigenvalues,eigenvectors and the spectral decomposition of the second-order symmetric tensor.