C-eigenvalue inclusion theorems for piezoelectric-type tensors (Q1726559)

Let \(\mathcal A=(a_{ijk})\) be an \(n\times n\times n\) real tensor. If \(a_{ijk}=a_{ikj}\) for all values of indices, then \(\mathcal A\) is called piezoelectric-type tensor. Let \(\mathcal C=(c_{ijk})\) be another \(n\times n\times n\) real tensor. If there exist a real scalar \(\lambda\) and real norm one \(n\times 1\) vectors \(x\) and \(y\) such that \(\mathcal A yy =\lambda x\), \(x\mathcal A y =\lambda y\), where \(\mathcal A yy\) and \(x\mathcal A y \) denote vectors with the entries, \(\sum_{i,k}c_{ijk}y_iy_k\) and \(\sum_{k,j}c_{kji}x_ky_j\), respectively, then \(\lambda\) is called a \(C\)-eigenvalue of \(\mathcal A\), and \(x\) and \(y\) are called associated left and right \(C\)-eigenvectors, respectively. The authors present three inclusion theorems to locate all \(C\)-eigenvalues for a given piezoelectric-type tensor.