Tensor Product in LaTeX (\otimes)

Where it is used

Circled times, the tensor product of vectors, spaces, modules or matrices (Kronecker product).

Multilinear algebra writes V \otimes W, quantum mechanics writes composite states |\psi\rangle \otimes |\phi\rangle, and matrix algebra uses A \otimes B for the Kronecker product. In machine learning it sometimes appears for outer products. It is a binary operator with medium spacing. For iterated products use the big operator \bigotimes_{i=1}^n V_i, and for the tensor product over a ring put the ring as a subscript, M \otimes_R N. Do not use \times, which is the Cartesian or cross product.

Also listed under: Linear Algebra, Topology & Abstract Math, Machine Learning & AI

Minimal compilable example

Paste this into an empty .tex file and compile it with pdfLaTeX, XeLaTeX or LuaLaTeX.

\documentclass{article}
\begin{document}
Tensor Product: $\otimes$

\[ (A \otimes B)(C \otimes D) = AC \otimes BD \]
\end{document}

Related symbols

• Circle Dot\odot
• Direct Sum\oplus
• Circle Minus\ominus
• Circle Slash\oslash
• Star\star
• Asterisk\ast

Frequently asked questions

How do I write Tensor Product in LaTeX?

Type \otimes in math mode, for example $(A \otimes B)(C \otimes D) = AC \otimes BD$. It works in plain LaTeX without loading any package.

Which package do I need for \otimes?

None. \otimes is part of standard LaTeX, so it compiles with nothing in the preamble but the document class.

Can I use \otimes in text mode, or only in math mode?

Only in math mode. Outside $...$ LaTeX stops with "Missing $ inserted", so write $\otimes$ in running text. If you only need the glyph in text, paste the Unicode character ⊗ instead (with XeLaTeX or LuaLaTeX and a font that has it).