Direct sum (⊕) and tensor (⊗) in LaTeX
\oplus and \otimes give the circled plus (⊕) and circled times (⊗), used for the direct sum of vector spaces or groups and the tensor product respectively.
LaTeX command
\oplusNot the symbol you meant? Draw it to find the command →
The direct sum combines two vector spaces or groups so that dimensions (or orders) add, and every element is a pair, one from each summand. It appears constantly in linear algebra when decomposing a space into independent pieces, for instance splitting a vector space into a subspace and its complement.
The tensor product is a different construction that builds a larger space from two smaller ones by combining, rather than pairing, their elements. It is central to multilinear algebra and to quantum mechanics, where the joint state space of two systems is \mathcal{H}_A \otimes \mathcal{H}_B, never \oplus.
\oplus is also widely reused outside pure algebra: in computer science and coding theory it commonly denotes bitwise XOR. \bigoplus and \bigotimes give the 'big operator' forms, analogous to \sum and \prod, for a direct sum or tensor product taken over an index.
Other forms
\otimestensor product (⊗)\odotcircled dot (⊙)Examples
$V \oplus W$the direct sum of two vector spaces
$u \otimes v$a tensor product of two vectors
$\mathbb{R}^2 \oplus \mathbb{R}^3 \cong \mathbb{R}^5$dimensions add under a direct sum
$\bigoplus_{i=1}^{n} V_i$a direct sum taken over an index
Tips and common mistakes
\oplusand\otimesneed no extra package, both are built into LaTeX.- In quantum mechanics, joint systems use
\otimes, not\oplus; a direct sum would describe a different, non-composite structure. \odot(a circled dot) is a related but less common symbol, sometimes used for the elementwise (Hadamard) product; keep it distinct from\otimes.- In computer science and coding theory
\oplususually means bitwise XOR rather than a direct sum, so state your convention if the audience might expect the algebra reading. - Use
\bigoplus_{i}and\bigotimes_{i}for sums or products over an index, rather than chaining many\oplusor\otimessymbols by hand.
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Convert your maths to LaTeX →FAQ
What does \oplus mean in LaTeX?
Circled plus, ⊕. In algebra it usually means the direct sum of vector spaces or groups; in computer science it commonly means bitwise XOR instead.
What does \otimes mean?
Circled times, ⊗, the tensor product of vector spaces, matrices, or quantum states.
How do I write a direct sum over an index?
Use the big operator form, \bigoplus_{i=1}^{n} V_i, rather than repeating \oplus by hand.
Is \oplus the same as ordinary addition?
No, it denotes a specific algebraic construction (or XOR in computing), not plain addition; use + for ordinary addition of numbers.