lk,s-Singular values and spectral radius of partially symmetric rectangular tensors

The real rectangular tensors arise from the strong ellipticity condition problem in solid mechanics and the entanglement problem in quantum physics. In this paper, we first study properties of l(k,s)-singular values of real rectangular tensors. Then, a necessary and sufficient condition for the positive definiteness of partially symmetric rectangular tensors is given. Furthermore, we show that the weak Perron-Frobenius theorem for nonnegative partially symmetric rectangular tensor keeps valid under some new conditions and we prove a maximum property for the largest lk, s-singular values of nonnegative partially symmetric rectangular tensor. Finally, we prove that the largest l(k, s)-singular value of nonnegative weakly irreducible partially symmetric rectangular tensor is still geometrically simple.