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An eigenvalue localization set for tensors and its applications.

Jianxing Zhao1, Caili Sang1

  • 1College of Data Science and Information Engineering, Guizhou Minzu University, Guiyang, Guizhou 550025 P.R. China.

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|March 25, 2017
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Summary
This summary is machine-generated.

Researchers developed a novel eigenvalue localization set for tensors, offering tighter bounds than previous methods. This advancement provides sharper minimum eigenvalue estimates for [Formula: see text]-tensors without complex subset selections.

Keywords:
[Formula: see text]-tensorslocalization setminimum eigenvaluenonnegative tensors

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Area of Science:

  • Numerical Analysis
  • Linear Algebra
  • Tensor Theory

Background:

  • Existing eigenvalue localization sets for tensors have limitations in tightness.
  • Previous methods for bounding tensor eigenvalues, such as those by Li et al. and Huang et al., can be improved.

Purpose of the Study:

  • To introduce a new, tighter eigenvalue localization set for tensors.
  • To establish sharper bounds for the minimum eigenvalue of [Formula: see text]-tensors.
  • To demonstrate the superiority of the new method over existing results, particularly Huang et al.'s, by avoiding the need for specific subset selections.

Main Methods:

  • Derivation of a new eigenvalue localization set for tensors.
  • Application of the new set to derive bounds for the minimum eigenvalue of [Formula: see text]-tensors.
  • Comparative analysis with existing methods (Li et al., Huang et al.).
  • Verification through numerical examples.

Main Results:

  • The new eigenvalue localization set is proven to be tighter than those previously reported.
  • New bounds for the minimum eigenvalue of [Formula: see text]-tensors are established and shown to be sharper.
  • The proposed method achieves tighter localization and sharper bounds without requiring the selection of specific proper subsets.

Conclusions:

  • The new eigenvalue localization set offers significant improvements in tensor analysis.
  • The derived bounds for [Formula: see text]-tensors are more precise than current state-of-the-art results.
  • The method's efficiency is demonstrated by its ability to yield better results without complex parameter choices.