14.2.6 Tensor Product Map Formula Area

The Tensor Product Map Formula Area explains how tensor products define mappings in algebra, essential for multilinear operations and tensor algebra.

Tensor Product Map Formula Area is the branch of study concerned with collecting and organizing the explicit defining formulas that specify how a tensor product of linear maps acts on elementary tensors, sums of elementary tensors, and coordinate representations, serving as the computational reference underlying the more conceptual treatments of the construction.

The Defining Formula on Elementary Tensors

Basic Statement

For linear maps

f : V_{1} \to W_{1}

g : V_{2} \to W_{2}

the formula area records the single defining rule

(f \otimes g) (v \otimes w) = f (v) \otimes g (w)

as the seed from which every other formula in this area is derived, either directly by substitution or by combination with linearity.

Formula for Sums of Elementary Tensors

Since a general element of V1 tensor V2 is a finite sum of elementary tensors, the formula area records the extended rule

(f \otimes g) (\sum_{i = 1}^{k} v_{i} \otimes w_{i}) = \sum_{i = 1}^{k} f (v_{i}) \otimes g (w_{i}),

which is the formula actually used in practice, since arbitrary elements of a tensor product are rarely presented as single elementary tensors.

Coordinate Formulas

Basis Expansion

Fixing bases of V1 and V2 with respect to which f and g have matrix entries a-i-j and b-k-l respectively, the formula area records the coordinate expansion

(f \otimes g) (e_{i} \otimes e_{k}) = (\sum_{j} a_{ji} f_{j}) \otimes (\sum_{l} b_{lk} g_{l}),

where f-j and g-l denote basis vectors of the target spaces, giving the exact coefficient of each basis elementary tensor in the image of a given basis elementary tensor of the domain.

Kronecker Product Formula

Collecting the coordinate expansion above into a single matrix equation gives the Kronecker product formula

[f \otimes g] = [f] \otimes [g],

where square brackets denote the matrix of a map with respect to the chosen bases, expressing the matrix of the tensor product map directly in terms of the matrices of f and g without further computation.

Formulas Governing Algebraic Properties

Composition Formula

The formula area records the composition rule

(f^{'} \otimes g^{'}) \circ (f \otimes g) = (f^{'} \circ f) \otimes (g^{'} \circ g)

alongside the linearity formulas

(f_{1} + f_{2}) \otimes g = (f_{1} \otimes g) + (f_{2} \otimes g),

(λ f) \otimes g = λ (f \otimes g) = f \otimes (λ g),

used together to reduce complicated expressions built from sums, scalar multiples, and composites of tensor products of maps to a normal form of a single tensor product of a composite on each factor.

Trace, Determinant, and Rank Formulas

For endomorphisms f and g of finite-dimensional spaces of dimensions n1 and n2, the formula area records

tr (f \otimes g) = tr (f) \cdot tr (g),

det (f \otimes g) = det (f)^{n_{2}} \cdot det (g)^{n_{1}},

rank (f \otimes g) = rank (f) \cdot rank (g),

as the three standard invariant formulas most frequently used when evaluating properties of a tensor product map without constructing it explicitly.

Formulas for Special Constructions

Identity Extension Formula

The formula area records the two single-factor extension formulas

(f \otimes {id}_{W}) (v \otimes w) = f (v) \otimes w,

({id}_{V} \otimes g) (v \otimes w) = v \otimes g (w),

which combine, via the composition formula, into the factorization

f \otimes g = (f \otimes {id}_{W}) \circ ({id}_{V} \otimes g),

serving as the formula most often used to reduce a general tensor product of maps to two simpler single-factor operations applied in sequence.