Dyadic product

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In mathematics, a dyadic product of two vectors is a third vector product next to dot product and cross product. The dyadic product is a square matrix that represents a tensor with respect to the same system of axes as to which the components of the vectors are defined that constitute the dyadic product. Thus, if

<math>

\mathbf{a} = \begin{pmatrix} a_x\\a_y\\a_z \end{pmatrix} \quad\hbox{and}\quad \mathbf{b} = \begin{pmatrix} b_x\\b_y\\b_z \end{pmatrix}, </math> then the dyadic product is

<math>

\mathbf{a}\otimes\mathbf{b} \;\stackrel{\mathrm{def}}{=}\; \begin{pmatrix} a_x b_x & a_x b_y & a_x b_z \\ a_y b_x & a_y b_y & a_y b_z \\ a_z b_x & a_z b_y & a_z b_z \\ \end{pmatrix} . </math> Sometimes it is useful to write a dyadic product as matrix product of two matrices, the first being a column matrix and the second a row matrix,

<math>

\mathbf{a}\otimes\mathbf{b} = \begin{pmatrix} a_x\\ a_y \\ a_z \end{pmatrix} \left(b_x,\;b_y,\; b_z\right) </math>

Example

<math>

\mathbf{a} = \begin{pmatrix} -1\\3\\2 \end{pmatrix} \quad\hbox{and}\quad \mathbf{b} = \begin{pmatrix} 5\\-3\\4 \end{pmatrix} \Longrightarrow \mathbf{a}\otimes\mathbf{b} = \begin{pmatrix} -5 & 3 & -4 \\ 15 & -9 & 12 \\ 10 & -6 & 8 \\ \end{pmatrix} </math>

Use

An important use of a dyadic product is the reformulation of a vector expression as a matrix-vector expression, for instance,

<math>

\mathbf{a}\, (\mathbf{b}\cdot\mathbf{c}) = (\mathbf{a}\otimes\mathbf{b})\, \mathbf{c}. </math> Indeed, take the ith component,

<math>

a_i (\mathbf{b}\cdot\mathbf{c}) = a_i \sum_{j=x,y,z} b_j c_j = \sum_{j=x,y,z} a_i b_j c_j = \sum_{j=x,y,z} (\mathbf{a}\otimes \mathbf{b})_{ij}\, c_j \quad\hbox{for}\quad i=x,y,z. </math> Or, equivalently, by use of the associative law valid for matrix multiplication,

<math>

(\mathbf{a}\otimes\mathbf{b})\, \mathbf{c} = \begin{pmatrix} a_x\\ a_y \\ a_z \end{pmatrix} \left(b_x,\;b_y,\; b_z\right) \begin{pmatrix}c_x\\ c_y\\ c_z\end{pmatrix} = \mathbf{a} (\mathbf{b}\cdot\mathbf{c}) </math>

Multiplication

The matrix multiplication of two dyadic products is given by,

<math>

(\mathbf{a}\otimes\mathbf{b})\, (\mathbf{c}\otimes\mathbf{d})= \begin{pmatrix} a_x\\ a_y \\ a_z \end{pmatrix} \left(b_x,\;b_y,\; b_z\right) \begin{pmatrix}c_x\\ c_y\\ c_z\end{pmatrix} \left(d_x,\;d_y,\; d_z\right) = (\mathbf{a}\otimes\mathbf{d})\, (\mathbf{b}\cdot\mathbf{c}). </math>

Generalization

In more general terms, a dyadic product is the representation of a simple element in (binary) tensor product space with respect to bases carrying the constituting spaces. Let U and V be linear spaces and UV be their tensor product space

<math>

u \in U, \quad v \in V\; \Longrightarrow\; u\otimes v \in U\otimes V. </math> If {ai} and {bj} are bases of U and V, respectively, then

<math>

u = \sum_{i=1}^m a_i u_i = (a_1,\; a_2,\;\ldots,\; a_m) \begin{pmatrix}u_1 \\u_2\\ \vdots\\ u_m\end{pmatrix} = (a_1,\; a_2,\;\ldots,\; a_m) \mathbf{u}, </math>

<math>

v = \sum_{j=1}^n b_j v_j = (b_1,\; b_2,\;\ldots,\; b_n) \begin{pmatrix}v_1 \\v_2\\ \vdots\\ v_n\end{pmatrix} = (b_1,\; b_2,\;\ldots,\; b_n) \mathbf{v} </math> and

<math>
u\otimes v = \sum_{i=1}^m\sum_{j=1}^n (a_i b_j) (u_i v_j)

= \sum_{i=1}^m\sum_{j=1}^n (a_i b_j) (\mathbf{u}\otimes \mathbf{v})_{ij} </math> The dyadic product uv is an m × n matrix that represents the simple tensor uv in UV.