Pre-order between diagonal elements and singular values of real matrices

Thompson obtained necessary and sufficient conditions on the existence of a real square matrix with prescribed diagonal elements, prescribed singular values and non-negative determinant. This result gave a Thompson's type pre-order on . In this paper, a new pre-order on motivated by two special classes of real matrices called pinching matrices and mirror sliding matrices will be introduced. We show that this pre-order is equivalent to Thompson's type pre-order. We further apply it to study convexity and inclusion relations on linear images of the special orthogonal orbits of real matrices. Moreover, inclusion relations among matrix sets motivated by Thompson's type pre-order, pinching matrices, mirror sliding matrices and orthostochastic matrices will be studied.