Dyadic product

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In mathematics, in particular multilinear algebra, the dyadic product

\mathbb{P} = \mathbf{u}\otimes\mathbf{v}

of a column vector \mathbf{u} and a row vector \mathbf{v} is the tensor product of the vectors. The result is a tensor of rank two (a matrix). It is a special case of the tensor product or Kronecker product, for vectors of the same dimension.

[edit] Example

\mathbf{u} \otimes \mathbf{v} = \begin{bmatrix} u_1 \\ u_2 \\ u_3 \end{bmatrix} \otimes \begin{bmatrix} v_1 & v_2 & v_3 \end{bmatrix} = \begin{bmatrix} u_1v_1 & u_1v_2 & u_1v_3 \\ u_2v_1 & u_2v_2 & u_2v_3 \\ u_3v_1 & u_3v_2 & u_3v_3 \end{bmatrix}.

[edit] Definition

Using Einstein's summation convention, the dyadic product

\mathbf{u} \otimes \mathbf{v}

may be defined by

\mathbb{P}_{ij} = u_i v_j.

Writing out the sums, this becomes

\sum_{i,j}u_i v_j \mathbf{e}_i \otimes \mathbf{e}_j^T.

[edit] See also

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