Editing Tensor (section)

=== Tensor product ===
{{Main|Tensor product}}

The [[tensor product]] takes two tensors, ''S'' and ''T'', and produces a new tensor, {{nowrap|{{math|''S'' ⊗ ''T''}}}}, whose order is the sum of the orders of the original tensors.  When described as multilinear maps, the tensor product simply multiplies the two tensors, i.e.,
<math display="block">(S \otimes T)(v_1, \ldots, v_n, v_{n+1}, \ldots, v_{n+m}) = S(v_1, \ldots, v_n)T(v_{n+1}, \ldots, v_{n+m}),</math>
which again produces a map that is linear in all its arguments.  On components, the effect is to multiply the components of the two input tensors pairwise, i.e.,
<math display="block">
  (S \otimes T)^{i_1\ldots i_l i_{l+1}\ldots i_{l+n}}_{j_1\ldots j_k j_{k+1}\ldots j_{k+m}} =
  S^{i_1\ldots i_l}_{j_1\ldots j_k} T^{i_{l+1}\ldots i_{l+n}}_{j_{k+1}\ldots j_{k+m}}.
</math>
If {{mvar|S}} is of type {{math|(''l'', ''k'')}} and {{mvar|T}} is of type {{math|(''n'', ''m'')}}, then the tensor product {{nowrap|{{math|''S'' ⊗ ''T''}}}} has type {{nowrap|{{math|(''l'' + ''n'', ''k'' + ''m'')}}}}.