Dyadic product

Dyadic product


In mathematics, in particular multilinear algebra, the dyadic product

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

of two vectors, \mathbf{u} and \mathbf{v}, each having the same dimension, is the tensor product of the vectors and results in a tensor of order two and rank one. It is also called outer product.

Contents

Components

With respect to a chosen basis \{\mathbf{e}_i\}, the components Pij of the dyadic product \mathbb{P} = \mathbf{u} \otimes \mathbf{v} may be defined by

\displaystyle P_{ij} = u_i v_j ,

where

\mathbf{u} = \sum_i u_i \mathbf{e}_i ,
\mathbf{v} = \sum_j v_j \mathbf{e}_j ,

and

\mathbb{P} = \sum_{i,j} P_{ij} \mathbf{e}_i \otimes \mathbf{e}_j .

Matrix representation

The dyadic product can be simply represented as the square matrix obtained by multiplying \mathbf{u} as a column vector by \mathbf{v} as a row vector. For example,


 \mathbf{u} \otimes \mathbf{v}
 \rightarrow
 \begin{bmatrix}
 u_1 \\
 u_2 \\
 u_3 \end{bmatrix}
 \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} ,

where the arrow indicates that this is only one particular representation of the dyadic product, referring to a particular basis. In this representation, the dyadic product is a special case of the Kronecker product.

Identities

The following identities are a direct consequence of the definition of the dyadic product[1]:


\begin{align}
  (\alpha \mathbf{u}) \otimes \mathbf{v} &= \mathbf{u} \otimes (\alpha \mathbf{v}) = \alpha (\mathbf{u} \otimes \mathbf{v}), \\
  \mathbf{u} \otimes (\mathbf{v} + \mathbf{w}) &= \mathbf{u} \otimes \mathbf{v} + \mathbf{u} \otimes \mathbf{w}, \\
  (\mathbf{u} + \mathbf{v}) \otimes \mathbf{w} &= \mathbf{u} \otimes \mathbf{w} + \mathbf{v} \otimes \mathbf{w}, \\
  (\mathbf{u} \otimes \mathbf{v}) \mathbf{w} &= \mathbf{u}\; (\mathbf{v} \cdot \mathbf{w}), \\ 
  \mathbf{u} \cdot (\mathbf{v} \otimes \mathbf{w}) &= (\mathbf{u} \cdot \mathbf{v})\; \mathbf{w}.
\end{align}

See also

Notes

  1. ^ See Spencer (1992), page 19.

References

  • A.J.M. Spencer (1992). Continuum Mechanics. Dover Publications. ISBN 0486435946. .

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