1010: Tensor Product of Tensors

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definition of tensor product of tensors

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Topics

About: vectors space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of tensors space with respect to field and $k$ vectors spaces and vectors space over field.

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Target Context

• The reader will have a definition of tensor product of tensors.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$F$: $\in \{\text{ the fields }\}$
$\{V_{1,1},...,V_{1,k_1},V_{2,1},...,V_{2,k_2}\}$: $\subseteq \{\text{ the }F\text{ vectors spaces }\}$
$\ast \otimes$: $:L(V_{1,1},...,V_{1,k_1}:F)\times L(V_{2,1},...,V_{2,k_2}:F)\to L(V_{1,1},...,V_{1,k_1},V_{2,1},...,V_{2,k_2}:F),(t^1,t^2)\mapsto t^1\otimes t^2$
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Conditions:
$\mathrm{\forall }{v}_{j,m}\in V_{j,m}(t^1\otimes t^2(v_{1,1},...,v_{1,k_1},v_{2,1},...,v_{2,k_2})=t^1(v_{1,1},...,v_{1,k_1})t^2(v_{2,1},...,v_{2,k_2}))$
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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 $:V_{1,1}\times ...\times V_{1,k_1}\times V_{2,1}\times ...\times V_{2,k_2}\to F$.

$t^1\otimes t^2(...,r{v}_{1,m}+r^{\prime }{v}_{1,m}^{\prime },...)=t^1(...,r{v}_{1,m}+r^{\prime }{v}_{1,m}^{\prime },...)t^2(...)=(r{t}^1(...,v_{1,m},...)+r^{\prime }{t}^1(...,v_{1,m}^{\prime },...))t^2(...)=r{t}^1(...,v_{1,m},...)t^2(...)+r^{\prime }{t}^1(...,v_{1,m}^{\prime },...)t^2(...)=r{t}^1\otimes t^2(...,v_{1,m},...)+r^{\prime }{t}^1\otimes t^2(...,v_{j,m}^{\prime },...)$.

$t^1\otimes t^2(...,r{v}_{2,m}+r^{\prime }{v}_{2,m}^{\prime },...)=t^1(...)t^2(...,r{v}_{2,m}+r^{\prime }{v}_{2,m}^{\prime },...)=t^1(...)(r{t}^2(...,v_{2,m},...)+r^{\prime }{t}^2(...,v_{2,m}^{\prime },...))=r{t}^1(...)t^2(...,v_{2,m},...)+r^{\prime }{t}^1(...)t^2(...,v_{2,m}^{\prime },...)=r{t}^1\otimes t^2(...,v_{2,m},...)+r^{\prime }{t}^1\otimes t^2(...,v_{2,m}^{\prime },...)$.

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 $t^j,t^{\prime j}\in L(V_{j,1},...,V_{j,k_j}:F)$ and each $r,r^{\prime }\in F$, $t^1\otimes ...\otimes (r{t}^j+r^{\prime }{t}^{\prime j})\otimes ...\otimes t^l=r{t}^1\otimes ...\otimes t^j\otimes ...\otimes t^l+r^{\prime }{t}^1\otimes ...\otimes t^{\prime j}\otimes ...\otimes t^l$: $t^1\otimes ...\otimes (r^{\prime }{t}^j+r^{\prime }{t}^{\prime j})\otimes ...\otimes t^l(v_{1,1},...,v_{l,k_l})=t^1(v_{1,1},...,v_{1,k_1})...(r{t}^j+r^{\prime }{t}^{\prime j})(v_{j,1},...,v_{j,k_j})...t^l(v_{l,1},...,v_{l,k_l}))=t^1(v_{1,1},...,v_{1,k_1})...(r{t}^j(v_{j,1},...,v_{j,k_j})+r^{\prime }{t}^{\prime j}(v_{j,1},...,v_{j,k_j}))...t^l(v_{l,1},...,v_{l,k_l}))=r{t}^1(v_{1,1},...,v_{1,k_1})...t^j(v_{j,1},...,v_{j,k_j})...t^l(v_{l,1},...,v_{l,k_l}))+r^{\prime }{t}^1(v_{1,1},...,v_{1,k_1})...t^{\prime j}(v_{j,1},...,v_{j,k_j})...t^l(v_{l,1},...,v_{l,k_l}))=(r{t}^1\otimes ...\otimes t^j\otimes ...\otimes t^l+r^{\prime }{t}^1\otimes ...\otimes t^{\prime j}\otimes ...\otimes t^l)(v_{1,1},...,v_{l,k_l})$.

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.

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References

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