A cubic hypermatrix of order $d$ can be considered as a structure matrix of a tensor with covariant order $r$ and contra-variant order $s=d-r$. Corresponding to this matrix expression of the hypermatrix, an eigenvector $x$ with respect to an eigenvalue $\lambda$ is proposed, called the universal eigenvector and eigenvalue of the hypermatrix. According to the action of tensors, if $x$ is decomposable, it is called a universal hyper-(UH-)eigenvector. Particularly, if all decomposed components are the same, $x$ is called a universal diagonal hyper (UDH-)eigenvector, which covers most of existing definitions of eigenvalue/eigenvector of hypermatrices. Using Semi-tensor product (STP) of matrices, the properties of universal eigenvalues/eigenvectors are investigated. Algorithms are developed to calculate universal eigenvalues/eigenvectors for hypermatrices. Particular efforts have been put on UDH- eigenvalues/eigenvectors, because they cover most of the existing eigenvalues/eigenvectors for hypermatrices. Some numerical examples are presented to illustrate that the proposed technique is universal and efficient.

翻译：一个$d$阶的超立方矩阵可被视为一个协变阶为$r$、逆变阶为$s=d-r$的张量的结构矩阵。针对该超矩阵的矩阵表达形式，本文提出一种关于特征值$\lambda$的特征向量$x$，称为超矩阵的通用特征向量与特征值。根据张量作用，若$x$可分解，则称其为通用超特征向量（UH-特征向量）。特别地，若所有分解分量相同，则$x$称为通用对角超特征向量（UDH-特征向量），其涵盖了现有超矩阵特征值/特征向量的大多数定义。利用矩阵的半张量积（STP）方法，研究了通用特征值/特征向量的性质，并开发了计算超矩阵通用特征值/特征向量的算法。本文重点研究了UDH-特征值/特征向量，因其涵盖现有超矩阵特征值/特征向量的大多数情形。最后通过数值算例验证了所提方法的普适性与高效性。