Trace of tensor product vs Tensor contraction

Question

I have come across various sources that talk about traces of tensors. How does that work? In particular, there seem to be such an equality:

$$ \text{Tr}(T_1\otimes T_2)=\text{Tr}(T_1)\text{Tr}(T_2)\;\;\;...(1) $$

as found in this Stackexchange question and this Wikipedia page.

The Wiki page "Tensor Contraction" speaks of tensor contraction as some generalization of trace, though without providing any formulation or example.

My questions: How do they all work? What is trace for a tensor? How does such trace interact with tensor product?

In particular, I have this contraction: (following Einstein's summation convention) $$ F^{\mu\nu}F_{\mu\nu} $$ where $F$ is a rank-2 tensor and each $F_{\mu\nu}$ is a $4\times 4$ matrix. Can it be expressed as trace of some sort? Subsequently can I apply (1) to split the expression into product of, say, the trace of $F$?

Additionally, $F^{\mu\nu}$ is anti-symmetric and I am trying to prove the above equals to zero. So being able to use (1) can be awesome.

Note: I have some although limited background in differential geometry and algebra. English words are great. But please supply formal definitions as well. At the same time, explanations with as little abstract algebraic constructions as possible would be much appreciated. Focus on finite dimension is fine. Extension to separable Hilbert space, partial trace and etc is welcomed too.

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EDIT: The second floor to in this post seems to be good. I don't understand, however, how tensor product of matrices work? Still, when does contraction come into play?

Answer 1

I try to answer starting from the case of square matrices. There is some care to take while considering a "hidden" isomorphism of vector spaces. In any case, let $V$ be a finite dim. vector spaces over a field $\mathbb K$ (for simplicity $\mathbb R$ ), with basis $\{e_i\}$ of cardinality $n$.

It is well known that there exists an isomorphism of vector spaces $$\Phi:\operatorname{Hom}_\mathbb K(V,V)\rightarrow V^{*}\otimes V, $$

with $$\Phi(\phi)=a_{ij}f_i\otimes e_j,$$ where $\phi\in \operatorname{Hom}_\mathbb K(V,V)$ and $\phi(e_i):=a_{ij}e_j$ for all $i,j=1,\dots,n$. $\{f_i\}$ is the dual basis on $V^{*}$ of the basis $\{e_i\}$ on $V$, i.e. $f_i(e_j)=\delta_{ij}$.

We use the Einstein convention for repeated indices.

We know how to define the trace operator $\operatorname{Tr}$ on the space $\operatorname{Hom}_\mathbb K(V,V)$; the trace is computed on the square matrix representing each linear map in $\operatorname{Hom}_\mathbb K(V,V)$. Let us move to the r.h.s. of the isomorphism $\Phi$.

• trace operator on $V^{*}\otimes V$

Let $$\operatorname{Tr}_1: V^{*}\otimes V\rightarrow \mathbb K, $$

be given by $\operatorname{Tr}_1(g\otimes v):=g(v)$.

Lemma $\operatorname{Tr}_1$ is linear and satisfies $$\operatorname{Tr}_1\circ \Phi=\operatorname{Tr}.$$

proof: just use definitions.

• trace operator on $(V^{*}\otimes V)\otimes\dots\otimes (V^{*}\otimes V) $

Using the $n=1$ case we introduce

$$\operatorname{Tr}_n: \underbrace{(V^{*}\otimes V)\otimes\dots\otimes (V^{*}\otimes V)}_{n-\text{times}} \rightarrow \mathbb K, $$

with $\operatorname{Tr}_n(f_1\otimes v_1\otimes\dots\otimes f_n\otimes v_n):=\prod_{i=1}^n f_i(v_i)$.

Lemma $\operatorname{Tr}_n$ is linear and invariant under permutations on $(V^{*}\otimes V)^{\otimes n}$; it satisfies $$\operatorname{Tr}_n\left(\Phi(\phi_1)\otimes\dots\otimes\Phi(\phi_n)\right)=\prod_{i=1}^n \operatorname{Tr}(\phi_i), $$ for all $\phi_i\in \operatorname{Hom}_\mathbb K(V,V)$.

proof: we prove the second statement. We introduce the notation $$\Phi(\phi_k):= a^k_{i_kj_k}f_{i_k}\otimes e_{i_k}\in V^{*}\otimes V,$$ for all $k=1,\dots,n$. We arrive at $$\operatorname{Tr}_n\left( (a^1_{i_1j_1}f_{i_1}\otimes e_{i_1})\otimes\dots\otimes (a^n_{i_nj_n}f_{i_n}\otimes e_{i_n})\right)=a^1_{i_1j_1}\dots a^n_{i_nj_n}f_{i_1}(e_{i_1})\dots f_{i_n}(e_{i_n})=\text{remember the definition of dual basis}= a^1_{i_1j_1}\dots a^n_{i_nj_n}\delta_{i_1j_1}\dots\delta_{i_nj_n}= a^1_{i_1i_1}\dots a^n_{i_ni_n}\\=\prod_{i=1}^n \operatorname{Tr}(\phi_i),$$ as claimed.