definition of tensor product of tensors
Topics
About: vectors space
The table of contents of this article
Starting Context
Target Context
- The reader will have a definition of tensor product of tensors.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\( F\): \(\in \{\text{ the fields }\}\)
\( \{V_{1, 1}, ..., V_{1, k_1}\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\)
...
\( \{V_{l, 1}, ..., V_{l, k_l}\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\)
\( L (V_{1, 1}, ..., V_{1, k_1}: F)\): \(= \text{ the tensors space }\)
...
\( L (V_{l, 1}, ..., V_{l, k_l}: F)\): \(= \text{ the tensors space }\)
\( f_1\): \(\in L (V_{1, 1}, ..., V_{1, k_1}: F)\)
...
\( f_l\): \(\in L (V_{l, 1}, ..., V_{l, k_l}: F)\)
\(*f_1 \otimes ... \otimes f_l\): \(\in L (V_{1, 1}, ..., V_{1, k_1}, ..., V_{l, 1}, ..., V_{l, k_l}: F)\)
//
Conditions:
\(\forall v_{j, m} \in V_{j, m} (f_1 \otimes ... \otimes f_l (v_{1, 1}, ..., v_{1, k_1}, ..., v_{l, 1}, ..., v_{l, k_l}) = f_1 (v_{1, 1}, ..., v_{1, k_1}) ... f_l (v_{l, 1}, ..., v_{l, k_l}))\)
//
2: Note
Each \(f_j\) need to be in \(L (V_{j, 1}, ..., V_{j, k_j}: F)\) instead of a general \(L (V_{j, 1}, ..., V_{j, k_j}: W)\), because otherwise, the multiplication,\(f_1 (v_{1, 1}, ..., v_{1, k_1}) ... f_l (v_{l, 1}, ..., v_{l, k_l})\), would not make sense.
Let us see that indeed \(f_1 \otimes ... \otimes f_l \in L (V_{1, 1}, ..., V_{1, k_1}, ..., V_{l, 1}, ..., V_{l, k_l}: F)\).
It is \(: V_{1, 1} \times ... \times V_{1, k_1} \times ... \times V_{l, 1} \times ... \times V_{l, k_l} \to F\).
\(f_1 \otimes ... \otimes f_l (..., r v_{j, m} + r' v'_{j, m}, ...) = f_1 (...) ... f_j (..., r v_{j, m} + r' v'_{j, m}, ...) ... f_l (...) = f_1 (...) ... (r f_j (..., v_{j, m}, ...) + r' f_j (..., v'_{j, m}, ...)) ... f_l (...) = r f_1 (...) ... f_j (..., v_{j, m}, ...) ... f_l (...) + r' f_1 (...) ... f_j (..., v'_{j, m}, ...)) ... f_l (...) = r f_1 \otimes ... \otimes f_l (..., v_{j, m}, ...) + r' f_1 \otimes ... \otimes f_l (..., v'_{j, m}, ...)\).
Let us see a property of tensor product of tensors.
For each \(f_j, f'_j \in \in L (V_{j, 1}, ..., V_{j, k_j}: F)\) and each \(r, r' \in F\), \(f_1 \otimes ... \otimes (r f_j + r' f'_j) \otimes ... \otimes f_l = r f_1 \otimes ... \otimes f_j \otimes ... \otimes f_l + r' f_1 \otimes ... \otimes f'_j \otimes ... \otimes f_l\): \(f_1 \otimes ... \otimes (r' f_j + r' f'_j) \otimes ... \otimes f_l (v_{1, 1}, ..., v_{l, k_l}) = f_1 (v_{1, 1}, ..., v_{1, k_1}) ... (r f_j + r' f'_j) (v_{j, 1}, ..., v_{j, k_j}) ... f_l (v_{l, 1}, ..., v_{l, k_l})) = f_1 (v_{1, 1}, ..., v_{1, k_1}) ... (r f_j (v_{j, 1}, ..., v_{j, k_j}) + r' f'_j (v_{j, 1}, ..., v_{j, k_j})) ... f_l (v_{l, 1}, ..., v_{l, k_l})) = r f_1 (v_{1, 1}, ..., v_{1, k_1}) ... f_{j, 1} (v_{j, 1}, ..., v_{j, k_j}) ... f_l (v_{l, 1}, ..., v_{l, k_l})) + r' f_1 (v_{1, 1}, ..., v_{1, k_1}) ... f'_j (v_{j, 1}, ..., v_{j, k_j}) ... f_l (v_{l, 1}, ..., v_{l, k_l})) = (r f_1 \otimes ... \otimes f_j \otimes ... \otimes f_l + r' f_1 \otimes ... \otimes f'_j \otimes ... \otimes f_l) (v_{1, 1}, ..., v_{l, k_l})\).
Do not confuse this concept with the concept of tensor product of k vectors spaces, which is a product of spaces and is a vectors space, while this concept is a product of tensors and is a tensor.
