12.4.3 Tensor Subtraction Componentwise Rule

Tensor Subtraction Componentwise Rule is the operational rule specifying that the difference of two tensors of the same type is computed by subtracting their corresponding components individually, index position by index position, with each output component depending solely on the two matching input components at that same position.

Statement of the Rule

General Formula

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

Each fixed choice of index values across the upper and lower index positions selects exactly one scalar component of $$D$$, computed by subtracting the corresponding scalar component of $$B$$ from that of $$A$$ at the same index choice.

Simple Rank Cases

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

For rank-2 tensors, the rule subtracts entries occupying the same row and column:

Relation to the Componentwise Addition Rule

Derivation

The componentwise subtraction rule is not an independent postulate but follows from applying the componentwise addition rule to $$A$$ and the negated tensor $$-B$$, whose components are the negatives of those of $$B$$. Adding $$A$$ to this negated tensor componentwise reproduces exactly the subtraction rule stated above.

No Independent Axiom Required

Because subtraction reduces cleanly to addition combined with negation, no separate componentwise axiom is required beyond the ones already governing addition and scalar multiplication.

Properties Following from the Rule

Non-Commutativity

Since ordinary scalar subtraction is not commutative, the componentwise tensor subtraction rule is likewise not commutative:

unless every component of $$A$$ equals the corresponding component of $$B$$.

Result of Subtracting a Tensor from Itself

Applying the rule with $$B$$ equal to $$A$$ yields a difference tensor whose every component is zero, since each scalar subtraction cancels exactly.

Basis Dependence and Transformation

Componentwise Subtraction Within a Fixed Basis

The rule as stated computes numerical differences of components relative to one chosen basis.

Consistency Across Bases

If both tensors are re-expressed in a new basis using the transformation law appropriate to their shared type, applying the componentwise subtraction rule to the new components produces exactly the components of the same difference tensor expressed in that new basis, since the transformation law is linear and therefore commutes with subtraction just as it does with addition.

Illustration