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In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors and vector spaces. In each case the significance of the symbol is the same: the most general bilinear operation.
A representative case is the Kronecker multiplication of any two rectangular arrays, considered as matrices.
Example:
Resultant rank = 2, resultant dimension = 12.
Here rank denotes the number of requisite indices, while dimension counts the number of degrees of freedom in the resulting array.
There is a general formula for the product of two (or more) tensors
We are assuming here orthogonal tensors, with no distinction of covariant and contravariant indices, for simplicity.
The parameters introduced above work out like this:
See also: Tensor-classical
With matrices this is usually called the Kronecker product, which is used to make clear that the result has a particular block structure imposed upon it, in which each element of the first matrix is replaced by the second matrix, scaled by that element. For matrices and this is:
Given multilinear maps and their tensor product is the multilinear function
The tensor product of two vector spaces V and W has a formal definition by the method of generators and relations. The equivalence class under these relations (given below) of is called a tensor and is denoted by . By construction, one can prove several identities between tensors and form an algebra of tensors.
To construct , take the vector space generated by and apply (factor out the subspace generated by) the following multilinear relations:
where are vectors from the appropriate spaces, and is from the underlying field.
We can then derive the identity , the zero in .
The resulting tensor product is itself a vector space, which can be verified by directly checking the vector space axioms. Given bases and for V and W respectively, the tensors of the form forms a basis for . The dimension of the tensor product therefore is the product of dimensions of the original spaces; for instance will have dimension .
If a, b, and c, are rank-one tensors (i.e. one-dimensional arrays), with indices i,j,k, respectively, then the tensor product of them is a rank-three tensor(i.e. three-dimensional array):
for( int i = 0; i < i_dim; i++) for( int j = 0; j < j_dim; j++) for( int k = 0; k < k_dim; k++) result[i][j][k] = a[i]*b[j]*c[k];If a is a rank-two tensor and b is a rank-one tensor, with indices i & j, and k, respectively, then the tensor product of them is a rank-three tensor:
for( int i = 0; i < i_dim; i++) for( int j = 0; j < j_dim; j++) for( int k = 0; k < k_dim; k++) result[i][j][k] = a[i][j]*b[k];The space of all bilinear maps from to is naturally isomorphic to the space of all linear maps from to . This is built into the construction; has only the relations that are necessary to ensure that a homomorphism from to will be bilinear.
The tensor product in fact satifies the universal property of being a fibered coproduct .
The tensor product of two Hilbert spaces is another Hilbert space, which is defined as described below.
Let H1 and H2 be two Hilbert spaces with inner products <·,·>1 and <·,·>2, respectively. Construct the tensor product of H1 and H2 as vector spaces as explained above. We can turn this vector space tensor product into an inner product space by defining
and extending by linearity. Finally, take the completion under this inner product. The result is the tensor product of H1 and H2 as Hilbert spaces.
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