In this paper we study multivariate polynomial functions in complex variables and their corresponding symmetric tensor representations. The focus is to find conditions under which such complex polynomials always take real values. We introduce the notion of symmetric conjugate forms and general conjugate forms, characterize the conditions for such complex polynomials to be real valued, and present their corresponding tensor representations. New notions of eigenvalues/ eigenvectors for complex tensors are introduced, extending similar properties from the Hermitian matrices. Moreover, we study a property of the symmetric tensors, namely, the largest eigenvalue (in the absolute value sense) of a real symmetric tensor is equal to its largest singular value; the result is also known as Banachs theorem. We show that a similar result holds for the complex case as well. Finally, we discuss some applications of the new notion of eigenvalues/eigenvectors for the complex tensors.
Bibliographical noteFunding Information:
Research Center for Management Science and Data Analytics, School of Information Management and Engineering, Shanghai University of Finance and Economics, Shanghai 200433, China (firstname.lastname@example.org). The work of this author was supported in part by the National Natural Science Foundation of China (grant 11401364). ?Department of Mathematics, University of Portsmouth, Portsmouth PO1 3HF, United Kingdom (email@example.com). ?Department of Industrial and Systems Engineering, University of Minnesota, Minneapolis, MN 55455 (firstname.lastname@example.org). The work of this author was supported in part by the National Science Foundation (grant CMMI-1462408).
- Banachs theorem
- Conjugate complex polynomial
- Symmetric complex tensor
- Tensor eigenvalue
- Tensor eigenvector