Dyadics

Dyadics are mathematical objects, representing linear functions of vectors. Dyadic notation was first established by Gibbs in 1884.

Definition

Dyadic A is formed by two vectors a and b (complex in general). Here, upper-case bold variables denote dyads whereas lower-case bold variables denote vectors.

$\mathbf {A} =\mathbf {a} \mathbf {b}$

In matrix form :

$\mathbf {A} ={\overset {\rightharpoonup }{a}}^{T}.{\overset {\rightharpoonup }{b}}=\left({\begin{array}{c}a_{1}\\a_{2}\end{array}}\right).\left({\begin{array}{cc}b_{1}&b_{2}\end{array}}\right)=\left({\begin{array}{cc}a_{1}b_{1}&a_{1}b_{2}\\a_{2}b_{1}&a_{2}b_{2}\end{array}}\right).$

In general algebraic form:

${\overset {=}{A}}=\sum _{i,j}a_{i}b_{j}{\overset {\rightharpoonup }{x_{i}}}{\overset {\rightharpoonup }{x_{j}}}$

where ${\overset {\rightharpoonup }{x_{i}}}$ are basics ( also known as coordinate axes) and i,j goes from 1 to space dimension, in general = 3.

A more general definition for multiple vectors ${\overset {\rightharpoonup }{a}}_{i},{\overset {\rightharpoonup }{b}}_{i}$

${\overset {=}{A}}=\sum _{i}{\overset {\rightharpoonup }{a}}_{i}{\overset {\rightharpoonup }{b}}_{i}={\overset {\rightharpoonup }{a}}_{1}{\overset {\rightharpoonup }{b}}_{1}+{\overset {\rightharpoonup }{a}}_{2}{\overset {\rightharpoonup }{b}}_{2}+{\overset {\rightharpoonup }{a}}_{3}{\overset {\rightharpoonup }{b}}_{3}+\cdots$

A dyadic which cannot be reduced to a sum of less than 3 dyads is said to be complete. In this case, the forming vectors are non-coplanar, see Chen (1983).

The following table classify dyadic:

	Determinant	Adjoint	Matrix
Zero	= 0	= 0	= 0
Linear	= 0	= 0	≠ 0 (single dyadic)
Planar	= 0	≠ 0 (single dyadic)	≠ 0
Complete	≠ 0	≠ 0	≠ 0

Dyadics algebra

Dyadic with vector

There are 4 operations for a vector with a dyadic

${\begin{aligned}\mathbf {c} \cdot \mathbf {a} \mathbf {b} &=\left(\mathbf {c} \cdot \mathbf {a} \right)\mathbf {b} \\\left(\mathbf {a} \mathbf {b} \right)\cdot \mathbf {c} &=\mathbf {a} \left(\mathbf {b} \cdot \mathbf {c} \right)\\\mathbf {c} \times \left(\mathbf {ab} \right)&=\left(\mathbf {c} \times \mathbf {a} \right)\mathbf {b} \\\left(\mathbf {ab} \right)\times \mathbf {c} &=\mathbf {a} \left(\mathbf {b} \times \mathbf {c} \right)\end{aligned}}$

Dyadic with dyadic

There are 5 operations for a dyadic to another dyadic:

Simple-dot product

$\left(\mathbf {ab} \right)\cdot \left(\mathbf {cd} \right)=\mathbf {a} \left(\mathbf {b} \cdot \mathbf {c} \right)\mathbf {d} =\left(\mathbf {b} \cdot \mathbf {c} \right)\mathbf {ad}$

For 2 general dyadics A and B:

${\overset {=}{A}}=\sum _{i}{\overset {\rightharpoonup }{a}}_{i}{\overset {\rightharpoonup }{b}}_{i}$

${\overset {=}{B}}=\sum _{j}{\overset {\rightharpoonup }{c}}_{j}{\overset {\rightharpoonup }{d}}_{j}$

${\overset {=}{A}}.{\overset {=}{B}}=\sum _{i,j}{\overset {\rightharpoonup }{a}}_{i}{\overset {\rightharpoonup }{b}}_{i}.{\overset {\rightharpoonup }{c}}_{j}{\overset {\rightharpoonup }{d}}_{j}=\sum _{i,j}\left({\overset {\rightharpoonup }{b}}_{i}.{\overset {\rightharpoonup }{c}}_{j}\right){\overset {\rightharpoonup }{a}}_{i}{\overset {\rightharpoonup }{d}}_{j}=\left({\overset {\rightharpoonup }{b}}_{1}.{\overset {\rightharpoonup }{c}}_{1}\right){\overset {\rightharpoonup }{a}}_{1}{\overset {\rightharpoonup }{d}}_{1}+\left({\overset {\rightharpoonup }{b}}_{1}.{\overset {\rightharpoonup }{c}}_{2}\right){\overset {\rightharpoonup }{a}}_{1}{\overset {\rightharpoonup }{d}}_{2}+{\text{...}}+\left({\overset {\rightharpoonup }{b}}_{2}.{\overset {\rightharpoonup }{c}}_{1}\right){\overset {\rightharpoonup }{a}}_{2}{\overset {\rightharpoonup }{d}}_{1}+\left({\overset {\rightharpoonup }{b}}_{2}.{\overset {\rightharpoonup }{c}}_{2}\right){\overset {\rightharpoonup }{a}}_{2}{\overset {\rightharpoonup }{d}}_{2}+{\text{...}}$

Double-dot product

There are two ways to define the double dot product. Many sources use a definition of the double dot product rooted in the matrix double-dot product,

$\mathbf {ab} \colon \mathbf {cd} =\left(\mathbf {a} \cdot \mathbf {d} \right)\left(\mathbf {b} \cdot \mathbf {c} \right)$

whereas other sources uses a definition unique (usually referred to as the "colon product") to Dyads:

$\left(\mathbf {ab} \right):\left(\mathbf {cd} \right)=\mathbf {c} \cdot \left(\mathbf {ab} \right)\cdot \mathbf {d} =\left(\mathbf {a} \cdot \mathbf {c} \right)\left(\mathbf {b} \cdot \mathbf {d} \right)$

One must be careful when deciding which convention to use. As there are no analogous matrix operations for the remaining dyadic products, no ambiguities in their definitions appear.

The double-dot product is commutative.

$\mathbf {A} \colon \!\mathbf {B} =\mathbf {B} \colon \!\mathbf {A}$

There is special double dot product with transpose

$\mathbf {A} \colon \!\mathbf {B} ^{\mathrm {T} }=\mathbf {A} ^{\mathrm {T} }\colon \!\mathbf {B}$

Another identity is:

${\overset {=}{A}}{\begin{array}{c}.\\.\end{array}}{\overset {=}{B}}=\left({\overset {=}{A}}.{\overset {=}{B}}^{T}\right){\begin{array}{c}.\\.\end{array}}{\overset {=}{I}}=\left({\overset {=}{B}}.{\overset {=}{A}}^{T}\right){\begin{array}{c}.\\.\end{array}}{\overset {=}{I}}$

Dot–cross product

$\left(\mathbf {ab} \right)\!\!\!{\begin{array}{c}_{\cdot }\\^{\times }\end{array}}\!\!\!\left(\mathbf {x} \mathbf {y} \right)=\left(\mathbf {a} \cdot \mathbf {x} \right)\left(\mathbf {b} \times \mathbf {y} \right)$

Cross–dot product

$\left(\mathbf {ab} \right)\!\!\!{\begin{array}{c}_{\times }\\^{\cdot }\end{array}}\!\!\!\left(\mathbf {xy} \right)=\left(\mathbf {a} \times \mathbf {x} \right)\left(\mathbf {b} \cdot \mathbf {y} \right)$

Double-cross product

$\left(\mathbf {ab} \right)\!\!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\!\left(\mathbf {xy} \right)=\left(\mathbf {a} \times \mathbf {x} \right)\left(\mathbf {b} \times \mathbf {y} \right)$

We can see that, for simple dyadics, that only formed by 2 vectors a and b,

$\mathbf {S} =\mathbf {a} \mathbf {b}$

Its double cross product is zero.

$\left(\mathbf {ab} \right)\!\!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\!\left(\mathbf {ab} \right)=\left(\mathbf {a} \times \mathbf {a} \right)\left(\mathbf {b} \times \mathbf {b} \right)=0$

However, for 2 general dyadics, there double-cross product is defined as:

${\overset {=}{A}}\!\!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\!{\overset {=}{B}}=\sum _{i,j}\left({\overset {\rightharpoonup }{a}}_{i}\times {\overset {\rightharpoonup }{c}}_{j}\right)\left({\overset {\rightharpoonup }{b}}_{i}\times {\overset {\rightharpoonup }{d}}_{j}\right)$

For double-cross product on itself, the result will not be zero. For a special dyadic A:

${\overset {=}{A}}=\sum _{i=1}^{3}{\overset {\rightharpoonup }{a}}_{i}{\overset {\rightharpoonup }{b}}_{i}$

${\overset {=}{A}}\!\!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\!{\overset {=}{A}}=\sum _{i,j}\left({\overset {\rightharpoonup }{a}}_{i}\times {\overset {\rightharpoonup }{a}}_{j}\right)\left({\overset {\rightharpoonup }{b}}_{i}\times {\overset {\rightharpoonup }{b}}_{j}\right)=\left({\overset {\rightharpoonup }{a}}_{1}\times {\overset {\rightharpoonup }{a}}_{2}\right)\left({\overset {\rightharpoonup }{b}}_{1}\times {\overset {\rightharpoonup }{b}}_{2}\right)+\left({\overset {\rightharpoonup }{a}}_{1}\times {\overset {\rightharpoonup }{a}}_{3}\right)\left({\overset {\rightharpoonup }{b}}_{1}\times {\overset {\rightharpoonup }{b}}_{3}\right)+\left({\overset {\rightharpoonup }{a}}_{2}\times {\overset {\rightharpoonup }{a}}_{1}\right)\left({\overset {\rightharpoonup }{b}}_{2}\times {\overset {\rightharpoonup }{b}}_{1}\right)+\left({\overset {\rightharpoonup }{a}}_{2}\times {\overset {\rightharpoonup }{a}}_{3}\right)\left({\overset {\rightharpoonup }{b}}_{2}\times {\overset {\rightharpoonup }{b}}_{3}\right)+\left({\overset {\rightharpoonup }{a}}_{3}\times {\overset {\rightharpoonup }{a}}_{1}\right)\left({\overset {\rightharpoonup }{b}}_{3}\times {\overset {\rightharpoonup }{b}}_{1}\right)+\left({\overset {\rightharpoonup }{a}}_{3}\times {\overset {\rightharpoonup }{a}}_{2}\right)\left({\overset {\rightharpoonup }{b}}_{3}\times {\overset {\rightharpoonup }{b}}_{2}\right)$

${\overset {=}{A}}\!\!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\!{\overset {=}{A}}=2\left\{\left({\overset {\rightharpoonup }{a}}_{1}\times {\overset {\rightharpoonup }{a}}_{2}\right)\left({\overset {\rightharpoonup }{b}}_{1}\times {\overset {\rightharpoonup }{b}}_{2}\right)+\left({\overset {\rightharpoonup }{a}}_{2}\times {\overset {\rightharpoonup }{a}}_{3}\right)\left({\overset {\rightharpoonup }{b}}_{2}\times {\overset {\rightharpoonup }{b}}_{3}\right)+\left({\overset {\rightharpoonup }{a}}_{3}\times {\overset {\rightharpoonup }{a}}_{1}\right)\left({\overset {\rightharpoonup }{b}}_{3}\times {\overset {\rightharpoonup }{b}}_{1}\right)\right\}$

Unit dyadic

For any vectors a, there exist a unit dyadic I, such that

$\mathbf {I} \cdot \mathbf {a} =\mathbf {a} \cdot \mathbf {I} =\mathbf {a}$

For any base of 3 vectors a, b and c, with reciprocal base ${\hat {\mathbf {a} }}$, ${\hat {\mathbf {b} }}$ and ${\hat {\mathbf {c} }}$, the unit dyadic defined by

$\mathbf {I} =\mathbf {a} {\hat {\mathbf {a} }}+\mathbf {b} {\hat {\mathbf {b} }}+\mathbf {c} {\hat {\mathbf {c} }}$

In Cartesian coordinates,

$\mathbf {I} =\mathbf {x} {\hat {\mathbf {x} }}+\mathbf {y} {\hat {\mathbf {y} }}+\mathbf {z} {\hat {\mathbf {z} }}$

For orthonormal base ${\overset {\rightharpoonup }{x_{i}}}={\overset {\rightharpoonup }{x_{i}'}}$,

$\mathbf {I} =\sum _{i}{\overset {\rightharpoonup }{x_{i}}}{\overset {\rightharpoonup }{x_{i}}}$

The corresponding matrix is

$\mathbf {I} =\left({\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\\\end{array}}\right)$

Rotation dyadic

For any vector a,

$\mathbf {a} \times \mathbf {I}$

is a 90 degree right hand rotation dyadic around a.

Some operations with unit dyadics

$\left(\mathbf {a} \times \mathbf {I} \right)\cdot \left(\mathbf {b} \times \mathbf {I} \right)=\mathbf {ab} -\left(\mathbf {a} \cdot \mathbf {b} \right)\mathbf {I}$

$\mathbf {I} \!\!{\begin{array}{c}_{\times }\\^{\cdot }\end{array}}\!\!\!\left(\mathbf {ab} \right)=\mathbf {b} \times \mathbf {a}$

$\mathbf {I} \!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\mathbf {A} =\left(\mathbf {A} \!\!{\begin{array}{c}_{\times }\\^{\times }\end{array}}\!\!\mathbf {I} \right)\mathbf {I} -\mathbf {A} ^{\mathrm {T} }$

$\mathbf {I} \;\colon \left(\mathbf {ab} \right)=\left(\mathbf {I} \cdot \mathbf {a} \right)\cdot \mathbf {b} =\mathbf {a} \cdot \mathbf {b} ={\text{Trace}}\left(\mathbf {ab} \right)$

See also

• Dyadic product
• Dyadic tensor

References

• ISMO V. Lindell (1992). Methods for Electromagnetic Field Analysis. ISBN 019856239. {{cite book}}: Check |isbn= value: length (help).
• Hollis C. Chen (1983). Theory of Electromagnetic Wave - A Coordinate-free approach. ISBN 0070106886..