Tensor product

In mathematics, the tensor product, denoted by otimes, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras and modules. In each case the significance of the symbol is the same: the most general bilinear operation.

Related Topics:
Mathematics - Vector - Matrices - Tensor - Algebras - Module - Bilinear operation

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

A representative case is the Kronecker product of any two rectangular arrays, considered as matrices.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Example:

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

egin{bmatrix}b_1 \ b_2 \ b_3 \ b_4end{bmatrix} otimes egin{bmatrix}a_1 & a_2 & a_3end{bmatrix} = egin{bmatrix}a_1b_1 & a_2b_1 & a_3b_1 \ a_1b_2 & a_2b_2 & a_3b_2 \ a_1b_3 & a_2b_3 & a_3b_3 \ a_1b_4 & a_2b_4 & a_3b_4end{bmatrix}

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Resultant rank = 2, resultant dimension = (4,3) = 12.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Here rank denotes the number of requisite indices, while dimension counts the number of degrees of freedom in the resulting array.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~