12.20.1 Tensor Addition Notation

Tensor Addition Notation provides a structured way to combine tensors, defining how their components interact under addition within the framework of tensor algebra.

Tensor Addition Notation is the specific symbolic convention used to write the sum of two or more tensors of matching variance type, encompassing the plain infix + symbol applied directly to tensor symbols, the corresponding coordinate-wise index notation in which the sum is written index by index, and the pointwise functional notation in which the sum is expressed as an equality of evaluated values on an arbitrary tuple of arguments. Because tensor addition is defined equivalently in several ways — as an abstract operation on multilinear maps, as an operation on coordinate arrays, and as a special case of the vector space structure on a fixed tensor type — its notation must clearly signal which of these equivalent perspectives a given expression is drawing on.

Direct Infix Notation

The Plain Sum Symbol

T + S

The most basic notation for tensor addition simply places the ordinary + symbol between two tensor symbols of the same type, exactly as with ordinary vector or scalar addition. This notation presupposes, without stating so explicitly, that T and S are already known to share the same variance type (p,q) and the same ambient space, since the + symbol alone carries no information distinguishing tensor addition from any other kind of addition.

Scalar-Weighted Sums

a T + b S

Combining tensor addition notation with scalar multiplication notation produces the standard linear-combination expression aT + bS, used throughout tensor algebra whenever the vector-space structure of a fixed tensor type is invoked, such as when confirming that a tensor operation is linear in one of its arguments.

Coordinate-Wise Index Notation

Componentwise Sum Written with Matching Indices

(T + S)^{i} = T^{i} + S^{i}

When tensors are expressed by their coordinates relative to a fixed basis, addition is written index by index, with the sum tensor's coordinate at each index position equal to the sum of the two input tensors' coordinates at that same position. This notation makes explicit that tensor addition, in coordinates, reduces to ordinary numerical addition applied independently to each matching pair of coordinates.

Requiring Identical Index Structure

(T + S)_{j}^{i} = T_{j}^{i} + S_{j}^{i}

For mixed tensors, the index notation for addition requires both terms to carry exactly the same pattern of upper and lower indices; writing Tⁱⱼ + Sⁱ with mismatched index patterns is not valid tensor addition notation, since it would attempt to combine a type-(1,1) tensor with a type-(1,0) tensor, violating the variance type compatibility that addition presupposes.

Pointwise Functional Notation

Equality of Evaluations as the Defining Notation

(T + S) (v_{1}, \dots, v_{k}) = T (v_{1}, \dots, v_{k}) + S (v_{1}, \dots, v_{k})

When tensors are treated as multilinear maps, this equation is the notation that actually defines the sum T + S as a new multilinear map, obtained by adding the two output values pointwise for every possible tuple of input vectors. This is the most fundamental of the three notational styles, since the coordinate-wise notation is derivable from it by substituting basis vectors for the arguments v₁,...,vₖ.

Diagram of the Three Notational Views of the Same Sum

Notational Requirements Inherited from Addition's Own Compatibility

Notation Must Not Obscure Type or Space Mismatch

Because tensor addition is only defined between tensors of matching variance type and ambient space, addition notation should never be applied to symbols whose type or space has not been established as matching; writing T + S for tensors of manifestly different type is a notational misuse that mirrors, at the level of symbols, the substantive type-incompatibility errors discussed for tensor addition closure and tensor operation type compatibility.

Parenthesization for Combined Operations

(T + S) \otimes U

When addition is combined with other operations, such as the tensor product, explicit parentheses are used to indicate that the addition is performed first, distinguishing (T + S) ⊗ U from any alternative reading in which the tensor product might otherwise be mistakenly applied only to one of the two addends before the sum is completed.

Consistency with Broader Tensor Operation Notation

Compatibility with Index and Summation Conventions

Tensor addition notation coexists directly with the summation convention and index-based operation notation used elsewhere; an expression such as TᵢSʲ + UᵢSʲ uses addition notation alongside implied-summation notation, and correctly parsing such an expression requires recognizing that the + binds the two addition terms together before any summation convention is applied to repeated indices within each term.

A Model for Notating Other Closed Operations

Because addition is the simplest tensor operation satisfying closure on a fixed type, its notation — a plain infix symbol requiring no further qualification about type or space, precisely because that type and space are held fixed by convention — serves as the notational template other operations diverge from whenever they are not similarly closed, explaining why operations like the tensor product or contraction require additional notational markers (⊗, explicit slot labels) that plain addition does not.