12.3.3 Tensor Addition Componentwise Rule

Tensor Addition Componentwise Rule is the operational rule specifying that the sum of two tensors of the same type is computed by adding their corresponding components individually, index position by index position, with no mixing or recombination between different components.

Statement of the Rule

General Formula

For two tensors $$A$$ and $$B$$ of type $$(p,q)$$, the componentwise rule defines the sum tensor $$C$$ by:

Each fixed choice of index values $$(i^1,\dots ,i^p,j_1,\dots ,j_q)$$ yields exactly one scalar component of $$C$$, obtained by adding the corresponding scalar components of $$A$$ and $$B$$ for that same index choice.

Simple Rank Cases

For vectors, treated as rank-1 tensors, the rule reduces to adding entries at matching positions:

For rank-2 tensors, such as matrices representing a linear map, the rule adds entries occupying the same row and column:

Properties Following from the Rule

Commutativity

Since ordinary scalar addition is commutative, the componentwise rule automatically inherits this property:

Associativity

Because scalar addition is associative, grouping three or more tensors for addition does not affect the result, so parentheses can be omitted entirely when summing several tensors of the same type.

Existence of a Zero Tensor

The componentwise rule implies the existence of a unique zero tensor of type $$(p,q)$$, all of whose components are zero, since adding it to any tensor leaves every component unchanged.

Existence of Additive Inverses

For any tensor $$A$$, the componentwise rule guarantees an additive inverse whose components are the negatives of the components of $$A$$, so that their sum yields the zero tensor at every index position.

Basis Dependence and Transformation

Componentwise Addition Within a Fixed Basis

The rule as stated operates within one chosen basis. The numerical values being added are the components relative to that basis.

Consistency Across Bases

If both tensors are re-expressed in a new basis using the appropriate transformation law, applying the componentwise addition rule to the new components produces exactly the components of the same sum tensor expressed in the new basis. This consistency is what distinguishes tensor addition from an arbitrary addition of number arrays, since the rule commutes with change of basis.

Illustration