12.6.1 Tensor Negation Additive Inverse

Tensor Negation Additive Inverse is the identification of the negation of a tensor, obtained by reversing the sign of every one of its components, as precisely the additive inverse of that tensor within the vector space of tensors of its type, meaning the unique tensor which, when added to the original, yields the zero tensor.

Defining Negation as an Inverse

Componentwise Negation

For a tensor $$A$$ of type $$(p,q)$$, its negation $$-A$$ is the tensor whose components are the negatives of the corresponding components of $$A$$:

Verifying the Inverse Property

Adding $$A$$ to $$-A$$ componentwise yields, at every index position, the sum of a number and its negative, which is always zero:

where $$0$$ denotes the zero tensor of type $$(p,q)$$, whose every component is zero. This confirms that $$-A$$ satisfies exactly the defining property of an additive inverse.

Relation to Scalar Multiplication

Negation as Multiplication by Negative One

Tensor negation can equivalently be obtained by scalar multiplication with the field element $$-1$$:

This equivalence follows from the algebraic laws governing scalar multiplication, since $$A+(-1)A=(1+(-1\left)\right)A=0A=0$$, matching the requirement for an additive inverse.

Uniqueness of the Additive Inverse

Only One Tensor Satisfies the Property

Within the vector space of tensors of type $$(p,q)$$, the additive inverse of any given tensor $$A$$ is unique. If two tensors $$B$$ and $$C$$ both satisfied $$A+B=0$$ and $$A+C=0$$, then adding $$-A$$ to both equations shows $$B=C$$, so the componentwise negation is the only tensor fulfilling the inverse property.

Consequence for Type Preservation

Since the additive inverse must itself be a tensor of the same type in order for the sum with $$A$$ to be defined at all, uniqueness confirms that negation, which manifestly preserves rank, variance, and dimension, is the only candidate that could serve this role.

Basis Independence of Negation

Consistency Under Change of Basis

Because negation is scalar multiplication by $$-1$$, and scalar multiplication commutes with the linear transformation law for change of basis, the negation of a tensor computed in one basis and then transformed to another basis matches the negation computed directly from the transformed tensor. The additive inverse is therefore an intrinsic property of the abstract tensor, not an artifact of a particular coordinate description.

Illustration