The extensions of the Riemannian structure include the Finslerian one, which provided in recent years successful models in various , elds like Biology, Physics, GTR, Monolayer Nanotechnology and Geometry of Big data. The present article provides the necessary notions on tensor Spectral data and on the HO-SVD and the Candecomp tensor decompositions, and further study several aspects related to the Spectral theory of the main symmetric Finsler tensors, the fundamental and the Cartan tensor. In particular, are addressed two Finsler models used in Langmuir-Blodgett Nanotechnology and in Oncology. As well, the HO-SVD and Candecomp decompositions are exempli, ed for these models and metric extensions of the eigenproblem are proposed.