12.5.1 Tensor Scalar Multiplication Field Element

Tensor Scalar Multiplication Field Element is the identification of the scalar used in tensor scalar multiplication as an element drawn from the same field over which the underlying vector space of the tensor is defined, ensuring that the multiplication is algebraically meaningful and compatible with the vector space structure carrying the tensor.

The Role of the Field

Field Underlying the Vector Space

Every vector space is defined over a specific field, commonly the real numbers or the complex numbers, which supplies the scalars used in linear combinations of vectors. Since a tensor of type $$(p,q)$$ is built from a vector space $$V$$ and its dual $$V^{*}$$, the scalar used in tensor scalar multiplication must be an element of that same field, denoted here $$F$$.

Formal Requirement

For a scalar $$k\in F$$ and a tensor $$A$$ built over a vector space defined on the field $$F$$, the scalar multiple $$kA$$ is well defined precisely because $$k$$ belongs to the same field that governs the vector space operations underlying the tensor. An element from an unrelated field would not have an established multiplication rule with the components of the tensor.

Why Field Membership Matters

Compatibility with Componentwise Multiplication

Scalar multiplication of a tensor multiplies each component by $$k$$. Since each component of the tensor is itself an element of the field $$F$$, the product of a field element with another field element is guaranteed to exist and to again be an element of $$F$$, because fields are closed under multiplication. If $$k$$ came from a different field, this closure would not be guaranteed.

Preserving Field Structure of the Result

Because the field is closed under multiplication, the resulting tensor $$kA$$ has all of its components still lying in $$F$$, so the scaled tensor remains a valid tensor over the same vector space and the same field as the original.

Consequences for the Vector Space of Tensors

Field of Scalars for the Tensor Space

The set of all tensors of a fixed type $$(p,q)$$ over $$V$$ itself forms a vector space over the same field $$F$$ that underlies $$V$$. This is only possible because the scalar multiplication operation on tensors draws its scalars from $$F$$, matching the field of the original vector space exactly.

Consistency with Field Axioms

The field axioms, including distributivity, associativity of multiplication, and the existence of a multiplicative identity, are inherited directly by the scalar multiplication operation on tensors, since the scalars used are genuine elements of $$F$$ and obey these same axioms.

Examples of Field Choice

Real Field

When the underlying vector space is defined over the real numbers, tensor scalar multiplication uses real number scalars, and every component of the resulting tensor remains a real number.

Complex Field

When the underlying vector space is defined over the complex numbers, tensor scalar multiplication uses complex scalars, allowing components to be scaled and rotated in the complex plane, with the resulting tensor still having complex-valued components.

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