Dyadic product

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 U ⊗ V be their tensor product space

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

If {a_i} and {b_j} 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 u ⊗ v is an m × n matrix that represents the simple tensor u ⊗ v in U ⊗ V.