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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


\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},

then the dyadic product is


\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} .

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,


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

Example


\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}

Use

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


\mathbf{a}\, (\mathbf{b}\cdot\mathbf{c}) = (\mathbf{a}\otimes\mathbf{b})\, \mathbf{c}.

Indeed, take the ith component,


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.

Or, equivalently, by use of the associative law valid for matrix multiplication,


(\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})

Multiplication

The matrix multiplication of two dyadic products is given by,


(\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}).

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


u \in U, \quad v \in V\; \Longrightarrow\; u\otimes v \in U\otimes V.

If {ai} and {bj} are bases of U and V, respectively, then


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},

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}

and


 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}

The dyadic product uv is an m × n matrix that represents the simple tensor uv in UV.

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