The Spectral Theory of Tensors (Rough Version)

The spectral theory of tensors is an important part of numerical multi-linear algebra, or tensor computation [48, 76, 90]. It is possible that the ideas of eigenvalues of tensors had been raised earlier. However, it was in 2005, the papers of Lim and Qi initiated the rapid developments of the spectral theory of tensors. In 2005, independently, Lim [57] and Qi [70] defined eigenvalues and eigenvectors of a real symmetric tensor, and explored their practical application in deter- mining positive definiteness of an even degree multivariate form. This work extended the classical concept of eigenvalues of square matrices, forms an important part of numerical multi-linear algebra, and has found applications or links with automatic control, statisti- cal data analysis, optimization, magnetic resonance imaging, solid mechanics, quantum physics, higher order Markov chains, spectral hypergraph theory, Finsler geometry, etc, and attracted attention of mathematicians from different disciplines. After six years’ developments, we may classify the spectral theory of tensors to 18 research topics. Be- fore describing these 18 research topics in four groups, we state the basic definitions and properties of eigenvalues of tensors. https://arxiv.org/abs/1201.3424v1