Conjugate Partial-Symmetric Tensors

Abstract. We study conjugate partial-symmetric (CPS) tensors in this paper. The basis and dimension of the CPS tensor space is discussed and the conjugate tensor product decomposition of a CPS tensor is obtained from the CPS identifying matrix. We also introduce the conjugate partial-symmetric positive semidefinite (CPSPSD) tensors and the conjugate partial-symmetric completely positive (CPSCP) tensors in the CPS tensor space. The properties of the cone of CPSPSD tensors are given and the CPS[Formula: see text]-eigenvalues and sum of squares representations of a CPSPSD tensor are investigated. In addition, we propose a semidefinite relaxation method to compute the smallest CPS[Formula: see text]-eigenvalue. Further, we show that a CPSCP tensor has a CPSCP decomposition, where all real and imaginary parts of the decomposition vectors are nonnegative. A moment sum of squares (moment-SOS) relaxation algorithm is proposed for detecting whether a CPS tensor is CPSCP or not. If it is, we give a CPSCP decomposition for it; if it is not, we obtain a certificate for that. Low-rank CPSCP decompositions are also discussed.