Tensor Product · aiwiki.page

The tensor product is a construction in linear algebra that combines two vector spaces over the same field into a new vector space, written $V\otimes_K W$ . Its defining feature is that bilinear relationships between vectors in the original spaces become linear relationships on the new space. Elements called pure tensors have the form $v\otimes w$ ; general elements are finite sums of these. The construction also extends to modules and to products of several spaces. (math.stonybrook.edu)

Definition and universal property

Let $V$ and $W$ be vector spaces over $K$ . Their tensor product consists of a vector space $V\otimes_K W$ together with a bilinear map

\tau:V\times W\longrightarrow V\otimes_K W, \qquad \tau(v,w)=v\otimes w.

Bilinearity means linearity in each argument separately while the other is fixed. This is the two-variable case of a multilinear map. (math.stonybrook.edu)

The pair satisfies a universal property: for every vector space $X$ and every bilinear map $b:V\times W\to X$ , there is exactly one linear map

\widetilde b:V\otimes_K W\longrightarrow X

such that $\widetilde b(v\otimes w)=b(v,w)$ . This characterizes the pair up to a unique compatible isomorphism, independently of coordinates or chosen bases. In particular, the dual space $(V\otimes_K W)^*$ corresponds naturally to the space of scalar-valued bilinear forms on $V\times W$ . (math.stonybrook.edu)

Construction and elementary relations

An explicit construction begins with the free vector space whose formal generators are all pairs $(v,w)$ . One then forms a quotient vector space by the subspace generated by expressions enforcing bilinearity. In the quotient, these relations become

\begin{aligned} (v+v')\otimes w&=v\otimes w+v'\otimes w,\\ v\otimes(w+w')&=v\otimes w+v\otimes w',\\ (av)\otimes w&=a(v\otimes w)=v\otimes(aw). \end{aligned}

The images of the formal pairs generate the tensor product, so every element is a finite linear combination of pure tensors. Such expressions need not be unique. This quotient construction proves that the object specified by the universal property exists. (math.stonybrook.edu)

Bases, dimensions, and decomposability

If $\{e_i\}$ is a basis of $V$ and $\{f_j\}$ is a basis of $W$ , then $\{e_i\otimes f_j\}$ is a basis of their tensor product. Consequently, for finite-dimensional spaces,

\dim(V\otimes_K W)=\dim(V)\dim(W).

Thus the tensor product of a two-dimensional and a three-dimensional space is six-dimensional. Expanding $v=\sum_i v_i e_i$ and $w=\sum_j w_j f_j$ gives

v\otimes w=\sum_{i,j}v_iw_j(e_i\otimes f_j).

The coefficients of a pure tensor therefore factor into products of coefficients from its two factors. (hitoshi.berkeley.edu)

Not every tensor admits this factorization. For example, with independent basis vectors in both spaces,

e_1\otimes f_1+e_2\otimes f_2

cannot be written as one pure tensor. To see this in coordinates, identify $K^m\otimes_K K^n$ with the space of $m\times n$ matrices by sending $e_i\otimes f_j$ to the corresponding matrix unit. A pure tensor becomes the outer-product matrix $vw^{\mathsf T}$ , whose rank is at most one; the displayed example has rank two. This provides a coordinate test for decomposability of two-factor tensors. (hitoshi.berkeley.edu)

Tensor products of maps and structural identities

Maps $A:V\to V'$ and $B:W\to W'$ induce a map

A\otimes B:V\otimes W\longrightarrow V'\otimes W', \qquad (A\otimes B)(v\otimes w)=A(v)\otimes B(w).

In ordered product bases, its matrix is the Kronecker product: each entry $a_{ij}$ of the matrix of $A$ is replaced by the block $a_{ij}B$ . This operation differs from ordinary matrix multiplication, which represents composition rather than combining actions on separate factors. (math.ucsd.edu)

Tensor products have canonical structural isomorphisms

(U\otimes V)\otimes W\cong U\otimes(V\otimes W), \qquad V\otimes W\cong W\otimes V, \qquad K\otimes V\cong V.

The interchange map sends $v\otimes w$ to $w\otimes v$ ; it does not assert literal equality of differently ordered tensors. Associativity allows unambiguous notation for products of several spaces, whose universal property represents multilinear maps in the same way that a two-factor product represents bilinear maps. (stacks.math.columbia.edu)

Modules and Hilbert spaces

In abstract algebra, the analogous construction combines modules over a commutative ring $R$ . The product $M\otimes_R N$ represents $R$ -bilinear maps. The base ring matters: it determines which scalar relations are imposed, so it is included in the notation when ambiguity is possible. The construction retains associativity and distributivity over direct sums. (stacks.math.columbia.edu)

For Hilbert spaces, the algebraic tensor product carries an inner product determined on pure tensors by

\langle u\otimes v,u'\otimes v'\rangle =\langle u,u'\rangle\langle v,v'\rangle.

Completing this space in the induced norm produces the Hilbert-space tensor product. In infinite dimensions, this completion is an essential additional step: its elements need not be finite sums of pure tensors. Products of orthonormal basis vectors form an orthonormal basis of the completed space. (aalexan3.math.ncsu.edu)

Composite quantum systems

In quantum mechanics, the state space of a composite system is modeled by the tensor product of its component Hilbert spaces. For two qubits, the product space has dimension four. A pure state that factors as $\psi_A\otimes\psi_B$ is a product state; a pure state that cannot factor exhibits quantum entanglement. For independently prepared mixed states, the composite density matrix is $\rho_A\otimes\rho_B$ . General composite states need not have this product form. (learning.quantum.ibm.com)