14.19 Tensor Map Product Transformation Behavior

Tensor Map Product Transformation Behavior is the collection of structural laws describing how the assignment $(f,g)\mapsto f\otimes g$ interacts with the other basic operations available on linear maps, namely composition, the identity map, linear combination, and invertibility, and it is these laws, rather than any single component formula, that make the tensor product of maps a well-behaved operation compatible with the rest of linear algebra.

Behavior Under Composition

The Interchange Law

For linear maps $f:V\to V^{\prime }$, $f^{\prime }:V^{\prime }\to V^{\prime \prime }$, $g:W\to W^{\prime }$, and $g^{\prime }:W^{\prime }\to W^{\prime \prime }$, the tensor product respects composition in the sense that

so that composing the two pairs of maps separately and then tensoring the results agrees with tensoring first and then composing. This identity is not a coincidence of a chosen basis; it follows directly from checking both sides on a simple tensor $v\otimes w$, where each side reduces to $f^{\prime }\left(f\right(v\left)\right)\otimes g^{\prime }\left(g\right(w\left)\right)$.

Preservation of the Identity Map

Taking ${\mathrm{id}}_V$ and ${\mathrm{id}}_W$ to be the identity maps on $V$ and $W$, the tensor product satisfies

Together with the interchange law, this makes $(f,g)\mapsto f\otimes g$ a functor of two variables, sending pairs of composable maps to composable maps in a manner consistent with identities and composition in both arguments simultaneously.

Behavior Under Linear Combination

Bilinearity in Each Argument

For fixed $g$, the map $f\mapsto f\otimes g$ is linear, meaning

and symmetrically, for fixed $f$, the map $g\mapsto f\otimes g$ is linear as well, so that $(f,g)\mapsto f\otimes g$ is jointly bilinear as a map from the pair of spaces of linear maps into the space of linear maps on the tensor product.

Consequence: No Distributivity Over Sums in Both Slots at Once

Bilinearity does not imply that $(f_1+f_2)\otimes (g_1+g_2)$ equals a simple sum of the four cross terms without cross-multiplication; expanding correctly using bilinearity in both slots gives four separate tensor product terms,

none of which simplify further, so that the tensor product of a sum of maps is generically a genuinely larger object than either summand's own tensor product.

Behavior Under Invertibility

Inverse of a Tensor Product

If $f$ and $g$ are both invertible, then $f\otimes g$ is invertible, with inverse

which follows immediately from the interchange law applied to $(f^{-1}\circ f)\otimes (g^{-1}\circ g)$ and the preservation of the identity map.

Behavior Under Rank

The rank of $f\otimes g$ is the product of the individual ranks,

so that $f\otimes g$ is invertible exactly when both $f$ and $g$ are invertible, and it is the zero map exactly when at least one of $f$ or $g$ is the zero map.

Naturality Under Change of Basis

Consistency With Basis-Independent Statements

Every law above is stated purely in terms of the maps themselves, without reference to any coordinate system, so each remains valid verbatim after any change of basis on any of the spaces involved; only the component matrices appearing in a coordinate computation of these laws transform, via the Kronecker-product conjugation rule, while the identities relating the maps hold unconditionally.

Compatibility With Duality

For finite-dimensional spaces, transformation behavior extends to duals: the transpose, or dual map, of a tensor product of maps satisfies

showing that transposition, like inversion and composition, commutes with the tensor product construction rather than interacting with it in some more complicated way.