Let f be a function mapping from the n-dimensional real Euclidean space to the same space. The first term of its Taylor series, if any, is a constant vector. The second term is a linear function. The third term is a quadratic function determined a cubic tensor. The cubic tensor is an n by n by n cube filled with real numbers. My question concerns to the tensor and the quadratic function generated by the tensor. What does it do with the space? The only thing what I know from my own calculation is that a pair of eigenvalue and eigenvector may exist such that the eigenvector is an eigenvector with this length. However, any other vector of the halfline defined by the eigenvector is not an eigenvector. Does anybody know a book where the properties of tensors are discussed? If yes, please write which book is that.
Dear Béla Vizvári
Most (linear) algebra University text books introduce and discuss the properties of the tensor functor on modules over rings and as a special case on vector spaces over a field F (which are F-modules). Here I can mention some of those text books
S. Lang , Algebra
Hungerford , Algebra
I hope you will find what you need therin.
Good luck
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