New Criterions-Based <inline-formula><math display="inline"><semantics><mi mathvariant="script">H</mi></semantics></math></inline-formula>-Tensors for Testing the Positive Definiteness of Multivariate Homogeneous Forms

Abstract

Positive definite homogeneous multivariate forms play an important role in polynomial problems and medical imaging, and the definiteness of forms can be tested using structured tensors. In this paper, we state the equivalence between the positive definite multivariate forms and the corresponding tensors, and explain the connection between the positive definite tensors with H-tensors. Then, based on the notion of diagonally dominant tensors, some criteria for H-tensors are presented. Meanwhile, with these links, we provide an iterative algorithm to test the positive definiteness of multivariate homogeneous forms and prove its validity theoretically. The advantages of the obtained results are illustrated by some numerical examples.

Keywords

• homogeneous multivariate form
• positive definiteness
• ℋ-tensor
• diagonal dominance
• irreducible
• non-zero element chain