Tensor Product in LaTeX

The tensor product symbol ⊗ is \otimes: V \otimes W. For a product over many factors use the big operator \bigotimes_{i=1}^{n} V_i, and for the direct sum ⊕ use \oplus and \bigoplus. No package is needed.

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Circled operators

V \otimes W
$V \otimes W$
Tensor product ⊗, a binary operator.
\bigotimes_{i=1}^{n} V_i
$⨂_{i = 1}^{n} V_{i}$
Big tensor product with limits below and above in display.
V \oplus W
$V \oplus W$
Direct sum ⊕.
\bigoplus_{i \in I} V_i
$⨁_{i \in I} V_{i}$
Big direct sum.
A \odot B
$A ⊙ B$
Circled dot ⊙ (Hadamard product in many texts).
V^{\otimes n}
$V^{\otimes n}$
Tensor power: the operator inside a superscript.

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LaTeX

Templates

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Tensor product (⊗) examples

Each result below is drawn from the LaTeX under it. Copy the code, edit it on this page, or open it in the full editor.

Tensor product of spaces
$\dim (V \otimes W) = \dim V \cdot \dim W$
LaTeX
```
\dim (V \otimes W) = \dim V \cdot \dim W
```
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Pure tensor
$v \otimes w \in V \otimes W$
LaTeX
```
v \otimes w \in V \otimes W
```
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Over a ring
$M \otimes_{R} N$
LaTeX
```
M \otimes_R N
```
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Kronecker product

A \otimes B = (\begin{matrix} a_{11} B & a_{12} B \\ a_{21} B & a_{22} B \end{matrix})

LaTeX

A \otimes B = \begin{pmatrix} a_{11}B & a_{12}B \\ a_{21}B & a_{22}B \end{pmatrix}

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Direct sum decomposition
$V = ⨁_{i = 1}^{k} V_{i}$
LaTeX
```
V = \bigoplus_{i=1}^{k} V_i
```
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Tensor power
$T^{k} (V) = V^{\otimes k}$
LaTeX
```
T^{k}(V) = V^{\otimes k}
```
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\otimes or \bigotimes

\otimes sits between two factors like +. \bigotimes is a large operator like \sum: in a displayed equation its limits go below and above, inline they go to the side. Use the big form only when it has limits or ranges over a family.

Subscripts on ⊗

A subscript on \otimes (M \otimes_R N) is set close to the symbol. If you want the ring under the symbol instead, use \mathbin{\underset{R}{\otimes}}.

Common mistakes

\times is the Cartesian or cross product, a different construction.
Instead of
```
V \times W
```
Write
```
V \otimes W
```
The small operator does not take limits; they end up as plain scripts.
Instead of
```
\otimes_{i=1}^{n} V_i
```
Write
```
\bigotimes_{i=1}^{n} V_i
```
Inside a superscript use the binary form; the big operator is too large there.
Instead of
```
V^{\bigotimes n}
```
Write
```
V^{\otimes n}
```

In Overleaf, MathJax and Word

Overleaf and pdfLaTeX
\otimes, \bigotimes, \oplus and \bigoplus work without packages.
MathJax and KaTeX
MathJax and KaTeX support all four commands and \odot.
Microsoft Word
In a Word equation \otimes followed by Space inserts ⊗ and \oplus inserts ⊕.

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