Tensor-generated matrices and tensor H-eigenvalues distribution

In this work, we introduce the concept of tensor-generated matrices, which transform an $m$-order $n$-dimensional tensor into an $n$-dimensional square matrix by grouping corresponding elements. We demonstrate that if the tensor-generated matrix from tensor $\mathcal{A}$ is an $H$-matrix, then $\mathcal{A}$ must be an $H$-tensor. While classical eigenvalue distributions for matrices are well-established, they do not directly apply to tensor eigenvalues, such as Brauer's Ovals of Cassini sets, Ostrowski sets, and $S$-type inclusion sets. To overcome this challenge, we aim to convert high-order tensor $H$-eigenvalue localization into tensor-generated matrix eigenvalue localization. By establishing connections, we successfully obtain higher-order tensor $H$-eigenvalue distributions based on the subclass of $H$-matrices and matrix eigenvalue distributions. This approach enables us to extend existing matrix eigenvalue localization sets to higher-order tensor eigenvalues, resulting in modified versions of Brauer's Ovals of Cassini sets, Ostrowski sets, and $S$-type inclusion sets.