A new S-type eigenvalue inclusion set for tensors and its applications

Area of Science:

• Numerical Analysis
• Linear Algebra
• Tensor Computations

Background:

• Eigenvalue localization sets are crucial for analyzing tensor properties.
• Existing methods have limitations in sharpness and applicability.

Purpose of the Study:

• To derive a novel S-type eigenvalue localization set for tensors.
• To establish sharper bounds for tensor eigenvalues compared to prior work.
• To apply these findings to improve bounds for spectral radius and H-eigenvalues.

Main Methods:

• Partitioning the tensor space into a subset S and its complement.
• Developing a new S-type eigenvalue localization set based on this partition.
• Comparing the derived set and bounds with existing literature.

Main Results:

• A new, sharper S-type eigenvalue localization set for tensors is established.
• The new set provides tighter bounds than those by Qi (2005), Li et al. (2014, 2015).
• New, tighter bounds for the spectral radius of nonnegative tensors and minimum H-eigenvalues of M-tensors are derived.

Conclusions:

• The newly developed S-type eigenvalue localization set offers significant improvements in sharpness.
• The derived bounds for spectral radius and H-eigenvalues are superior to existing ones.
• This research contributes to more accurate tensor analysis and computation.