notes / Studies in Algebra and Group Theory / chapter 5: Multilinear Algebra / From my notebooks
N27 Tensor products, trace, and symmetric tensors
as a tensor product
where the second isomorphism sends .
In bases: take a basis of , the dual basis of , and a basis of . Then
Trace of a linear operator
The usual definition: the trace of a square matrix is
and
(read the lecture notes lol).
If is upper triangular with diagonal entries (these are the eigenvalues), then
where is the multiplicity of .
Basis-free: the trace is the composite
where the second map comes from the bilinear map
Given a basis of and the dual basis of , along we get
Multilinear maps
Definition N27.1. A map is multilinear if it is linear in each variable separately.
Symmetric tensors
Definition N27.2. is symmetric if for all .
Let meow . Just do it:
Finally, this is MUCH BETTER.