12.4.2 Tensor Subtraction Additive Inverse Use

Tensor Subtraction Additive Inverse Use is the technique of expressing tensor subtraction not as an independent primitive operation but as an application of tensor addition combined with the additive inverse of the tensor being subtracted, so that every subtraction is carried out by adding a negated tensor.

The Role of the Additive Inverse

Definition of the Additive Inverse

For a tensor $$B$$ of type $$(p,q)$$, its additive inverse, written $$-B$$, is the unique tensor of the same type whose components are the negatives of the corresponding components of $$B$$:

This additive inverse arises as a special case of scalar multiplication, obtained by multiplying every component of $$B$$ by the scalar $$-1$$.

Subtraction Constructed from Addition and Inversion

Rather than treating subtraction as a separate rule, it is constructed entirely from two already established operations:

This means that no new componentwise rule needs to be introduced for subtraction beyond the rules already governing addition and scalar multiplication.

Why This Construction Is Used

Economy of Definitions

By defining subtraction through the additive inverse, the theory of tensor algebra avoids introducing a redundant operation. Every property of subtraction can instead be derived from the already proven properties of addition and scalar multiplication, keeping the axiomatic foundation minimal.

Guaranteed Consistency

Since addition and scalar multiplication are both known to preserve tensor type and to behave consistently under changes of basis, expressing subtraction through them automatically inherits these guarantees, without requiring a separate proof specific to subtraction.

Verifying Key Properties Through This Construction

The Zero Result

Using the additive inverse construction, subtracting a tensor from itself gives:

which follows directly from the vector space axiom guaranteeing that every element has an additive inverse summing to the zero element.

Distribution Over Scalars

Because scalar multiplication distributes over tensor addition, and the additive inverse of $$B$$ is itself a scalar multiple of $$B$$, the additive inverse construction directly yields:

Practical Use in Computation

Two-Step Procedure

When computing the difference of two tensors, the additive inverse use suggests a two-step procedure: first negate every component of the second tensor, then apply the ordinary componentwise addition rule between the first tensor and this negated tensor.

Applicability Across All Tensor Types

Because negation via scalar multiplication is defined for tensors of any type, this two-step procedure applies uniformly regardless of the rank or variance of the tensors involved, as long as both tensors share the same type.

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