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:
\( F\): \(\in \{\text{ the fields }\}\)
\( \{V_{1, 1}, ..., V_{1, k_1}, V_{2, 1}, ..., V_{2, k_2}\}\): \(\subseteq \{\text{ the } F \text{ vectors spaces }\}\)
\(*\otimes\): \(: L (V_{1, 1}, ..., V_{1, k_1}: F) \times L (V_{2, 1}, ..., V_{2, k_2}: F) \to L (V_{1, 1}, ..., V_{1, k_1}, V_{2, 1}, ..., V_{2, k_2}: F), (t^1, t^2) \mapsto t^1 \otimes t^2\)
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Conditions:
\(\forall v_{j, m} \in V_{j, m} (t^1 \otimes t^2 (v_{1, 1}, ..., v_{1, k_1}, v_{2, 1}, ..., v_{2, k_2}) = t^1 (v_{1, 1}, ..., v_{1, k_1}) t^2 (v_{2, 1}, ..., v_{2, k_2}))\)
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2: Note
Each \(t^j\) 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,\(t^1 (v_{1, 1}, ..., v_{1, k_1}) t^2 (v_{2, 1}, ..., v_{2, k_2})\), would not make sense.
Let us see that indeed \(t^1 \otimes t^2 \in L (V_{1, 1}, ..., V_{1, k_1}, V_{2, 1}, ..., V_{2, k_2}: F)\).
It is \(: V_{1, 1} \times ... \times V_{1, k_1} \times V_{2, 1} \times ... \times V_{2, k_2} \to F\).
\(t^1 \otimes t^2 (..., r v_{1, m} + r' v'_{1, m}, ...) = t^1 (..., r v_{1, m} + r' v'_{1, m}, ...) t^2 (...) = (r t^1 (..., v_{1, m}, ...) + r' t^1 (..., v'_{1, m}, ...)) t^2 (...) = r t^1 (..., v_{1, m}, ...) t^2 (...) + r' t^1 (..., v'_{1, m}, ...) t^2 (...) = r t^1 \otimes t^2 (..., v_{1, m}, ...) + r' t^1 \otimes t^2 (..., v'_{j, m}, ...)\).
\(t^1 \otimes t^2 (..., r v_{2, m} + r' v'_{2, m}, ...) = t^1 (...) t^2 (..., r v_{2, m} + r' v'_{2, m}, ...) = t^1 (...) (r t^2 (..., v_{2, m}, ...) + r' t^2 (..., v'_{2, m}, ...)) = r t^1 (...) t^2 (..., v_{2, m}, ...) + r' t^1 (...) t^2 (..., v'_{2, m}, ...) = r t^1 \otimes t^2 (..., v_{2, m}, ...) + r' t^1 \otimes t^2 (..., v'_{2, m}, ...)\).
Let us see that tensor product of tensors is associative.
\((t^1 \otimes t^2) \otimes t^3 (v_{1, 1}, ..., v_{1, k_1}, v_{2, 1}, ..., v_{2, k_2}, v_{3, 1}, ..., v_{3, k_3}) = (t^1 \otimes t^2) (v_{1, 1}, ..., v_{1, k_1}, v_{2, 1}, ..., v_{2, k_2}) t^3 (v_{3, 1}, ..., v_{3, k_3}) = t^1 (v_{1, 1}, ..., v_{1, k_1}) t^2 (v_{2, 1}, ..., v_{2, k_2}) t^3 (v_{3, 1}, ..., v_{3, k_3})\).
\(t^1 \otimes (t^2 \otimes t^3) (v_{1, 1}, ..., v_{1, k_1}, v_{2, 1}, ..., v_{2, k_2}, v_{3, 1}, ..., v_{3, k_3}) = t^1 (v_{1, 1}, ..., v_{1, k_1}) (t^2 \otimes t^3) (v_{2, 1}, ..., v_{2, k_2}, v_{3, 1}, ..., v_{3, k_3}) = t^1 (v_{1, 1}, ..., v_{1, k_1}) t^2 (v_{2, 1}, ..., v_{2, k_2}) t^3 (v_{3, 1}, ..., v_{3, k_3})\).
So, while \(t^1 \otimes ... \otimes t^n := (...((t^1 \otimes t^2) \otimes t^3) ... \otimes t^{n - 1}) \otimes t^n\), it can be associated in any way.
Let us see a property of tensor product of tensors.
For each \(t^j, t'^j \in L (V_{j, 1}, ..., V_{j, k_j}: F)\) and each \(r, r' \in F\), \(t^1 \otimes ... \otimes (r t^j + r' t'^j) \otimes ... \otimes t^l = r t^1 \otimes ... \otimes t^j \otimes ... \otimes t^l + r' t^1 \otimes ... \otimes t'^j \otimes ... \otimes t^l\): \(t^1 \otimes ... \otimes (r' t^j + r' t'^j) \otimes ... \otimes t^l (v_{1, 1}, ..., v_{l, k_l}) = t^1 (v_{1, 1}, ..., v_{1, k_1}) ... (r t^j + r' t'^j) (v_{j, 1}, ..., v_{j, k_j}) ... t^l (v_{l, 1}, ..., v_{l, k_l})) = t^1 (v_{1, 1}, ..., v_{1, k_1}) ... (r t^j (v_{j, 1}, ..., v_{j, k_j}) + r' t'^j (v_{j, 1}, ..., v_{j, k_j})) ... t^l (v_{l, 1}, ..., v_{l, k_l})) = r t^1 (v_{1, 1}, ..., v_{1, k_1}) ... t^j (v_{j, 1}, ..., v_{j, k_j}) ... t^l (v_{l, 1}, ..., v_{l, k_l})) + r' t^1 (v_{1, 1}, ..., v_{1, k_1}) ... t'^j (v_{j, 1}, ..., v_{j, k_j}) ... t^l (v_{l, 1}, ..., v_{l, k_l})) = (r t^1 \otimes ... \otimes t^j \otimes ... \otimes t^l + r' t^1 \otimes ... \otimes t'^j \otimes ... \otimes t^l) (v_{1, 1}, ..., v_{l, k_l})\).
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.
