Improvement of S-type localization sets of C-eigenvalues for piezoelectric-type tensors

The $C$-eigenvalues of piezoelectric-type tensors are crucial for understanding both the piezoelectric and converse piezoelectric effects. He et al. (2021) introduced an $S$-type inclusion set to locate all $C$-eigenvalues of such tensors. However, a straightforward counterexample demonstrates that this method can occasionally be inaccurate. In this paper, we propose an improved $S$-type localization sets for the $C$-eigenvalues of piezoelectric-type tensors. Additionally, we demonstrate that the proposed eigenvalue localization sets improve upon a recent result. Finally, numerical experiments are conducted to validate the effectiveness of our main results.