The decompositions and positive semidefiniteness of fourth-order conjugate partial-symmetric tensors with applications

TL;DR

This paper explores the properties, decompositions, and positive semidefiniteness of conjugate partial-symmetric tensors, extending matrix concepts to higher-order tensors with applications in real and complex cases.

Contribution

It introduces orthogonal matrix decompositions for CPS tensors and links tensor positive semidefiniteness to various decompositions, advancing tensor analysis methods.

Findings

Orthogonal matrix decomposition inherits properties from matrix decompositions.

Real CPS tensors differ subtly from complex cases in decomposition.

Positive semidefiniteness relates to tensor decompositions and nonnegativity.

Abstract

Conjugate partial-symmetric (CPS) tensor is a generalization of Hermitian matrices. For the CPS tensor decomposition some properties are presented. For real CPS tensors in particular, we note the subtle difference from the complex case of the decomposition. In addition to traditional decompositions in the form of the sum of rank-one tensors, we focus on the orthogonal matrix decomposition of CPS tensors, which inherits nice properties from decomposition of matrices. It then induces a procedure that reobtain the CPS decomposable property of CPS tensors. We also discuss the nonnegativity of the quartic real-valued symmetric conjugate form corresponding to fourth-order CPS tensors in real and complex cases, and establish its relationship to different positive semidefiniteness based on different decompositions. Finally, we give some examples to illustrate the applications of presented propositions.