14 Tensor Products of Maps

Tensor Products of Maps is a construction that extends linear maps between vector spaces to a linear map between their tensor products, assigning to every pair of linear maps

a single linear map

characterized on elementary tensors by

and extended to all of the tensor product by linearity.

Existence and Well-Definedness

The Bilinear Map Underlying the Construction

The assignment of elementary tensors alone does not define a linear map on the tensor product, since not every element of the tensor product is an elementary tensor, and elementary tensors satisfy nontrivial relations among themselves. The construction begins instead from the map

given by

Because tensor multiplication in the target space is bilinear and both $f$ and $g$ are linear, $\beta$ is bilinear in the pair $(v_1, v_2)$: it is additive and homogeneous in each argument separately.

Invocation of the Universal Property

The tensor product $V_1 \otimes V_2$ is characterized by a universal property: every bilinear map out of $V_1 \times V_2$ factors uniquely through the canonical bilinear map

Applying this property to the bilinear map $\beta$ produces a unique linear map $f \otimes g$ on $V_1 \otimes V_2$ satisfying $(f \otimes g) \circ \otimes = \beta$, which is precisely the defining formula on elementary tensors stated above. Uniqueness guarantees that the value of $f \otimes g$ on any element of the tensor product is fully determined once its values on elementary tensors are fixed, so the relations obeyed by elementary tensors are automatically respected.

Functorial Behavior

Compatibility with Composition

If $f' : W_1 \to U_1$ and $g' : W_2 \to U_2$ are further linear maps, then

Both sides agree on elementary tensors by direct substitution, and since elementary tensors span the tensor product, the two linear maps coincide everywhere. This identity shows that the tensor product of maps is compatible with composition in each factor independently.

Compatibility with Identity Maps

The tensor product of identity maps is again an identity map:

Together with the composition rule, this shows that pairing $(V, W) \mapsto V \otimes W$ on objects with $(f, g) \mapsto f \otimes g$ on morphisms defines a functor of two variables from the category of vector spaces to itself, covariant in each argument.

Linearity of the Assignment

For fixed $g$, the map sending $f$ to $f \otimes g$ is itself linear:

and symmetrically in the second argument. Consequently the map

is itself bilinear, and by the universal property it induces a canonical linear map

which is an isomorphism when all spaces involved are finite-dimensional.

Matrix Representation

The Kronecker Product

When $V_1, V_2, W_1, W_2$ are finite-dimensional with chosen bases, and $f$, $g$ are represented respectively by matrices $A$ of size $m_1 \times n_1$ and $B$ of size $m_2 \times n_2$, the map $f \otimes g$ is represented, in the induced basis of elementary tensors of the basis vectors, by the Kronecker product $A \otimes B$, an $m_1 m_2 \times n_1 n_2$ matrix whose block form is

This block structure is a direct consequence of the defining rule on elementary tensors: the coordinate vector of $(f \otimes g)(e_i \otimes e_j)$ is the column-by-column combination of the coordinates of $f(e_i)$ scaled by the coordinates of $g(e_j)$, which is exactly the pattern of the Kronecker product.

Determinant, Trace, and Eigenvalues

When $f$ and $g$ are endomorphisms of finite-dimensional spaces of dimensions $n_1$ and $n_2$, the induced endomorphism $f \otimes g$ satisfies

and if $\lambda$ is an eigenvalue of $f$ with eigenvector $u$, while $\mu$ is an eigenvalue of $g$ with eigenvector $w$, then $\lambda \mu$ is an eigenvalue of $f \otimes g$ with eigenvector $u \otimes w$, since

The complete spectrum of $f \otimes g$ is the set of all such pairwise products of eigenvalues of $f$ and of $g$, counted with the product of the corresponding multiplicities.

Structural Properties

Injectivity and Surjectivity

Over a field, if $f$ and $g$ are both injective, then $f \otimes g$ is injective; if $f$ and $g$ are both surjective, then $f \otimes g$ is surjective. In particular, if $f$ and $g$ are isomorphisms, then $f \otimes g$ is an isomorphism, with inverse

The converse implications require some care over general rings, but hold without restriction when the underlying scalars form a field and the spaces are finite-dimensional.

Rank

For finite-dimensional spaces, the rank of $f \otimes g$ equals the product of the ranks of $f$ and $g$:

This follows from the Kronecker product description together with the fact that a Kronecker product of two matrices of full rank restricted to their column spaces has rank equal to the product of the individual ranks, since row and column reductions on $A$ and $B$ propagate blockwise through $A \otimes B$.

Special Cases

Tensoring with the Identity

Fixing $g = \mathrm{id}_{V_2}$ produces the map

often used to extend a linear map to act "coordinatewise" on a tensor product with a fixed auxiliary space, for instance when extending scalars or applying a map to one leg of a tensor while leaving the other leg untouched.

Tensor Products of Dual Maps

If $f^{} : W_1^{} \to V_1^{}$ and $g^{} : W_2^{} \to V_2^{}$ denote the transposes (dual maps) of $f$ and $g$, then under the canonical identification of $(V_1 \otimes V_2)^{}$ with $V_1^{} \otimes V_2^{}$ in the finite-dimensional case, the dual of $f \otimes g$ corresponds to $f^{} \otimes g^{*}$, so transposition and tensoring of maps commute with one another.

Iterated and Multiple Tensor Products

The construction extends without modification to finite families of linear maps $f_1, \dots, f_k$ with $f_i : V_i \to W_i$, producing

acting on elementary tensors by $(f_1 \otimes \cdots \otimes f_k)(v_1 \otimes \cdots \otimes v_k) = f_1(v_1) \otimes \cdots \otimes f_k(v_k)$, with all of the compositional, matrix, and rank properties above generalizing componentwise.