Dyadics

Dyadic algebra

Product of dyadic and vector

There are four operations defined on a vector and dyadic, constructed from the products defined on vectors.

Product of dyadic and dyadic

There are five operations for a dyadic to another dyadic. Let a, b, c, d be real vectors. Then:

Letting

${\displaystyle \mathbf{A}=\sum _i{\mathbf{a}}_i{\mathbf{b}}_i,\mathbf{B}=\sum _j{\mathbf{c}}_j{\mathbf{d}}_j}$

be two general dyadics, we have:

Double-dot product

The first definition of the double-dot product is the Frobenius inner product,

${\displaystyle \begin{array}{cc}\mathrm{tr}\left(\mathbf{A}{\mathbf{B}}^{\mathsf{T}}\right)& =\sum _{i,j}\mathrm{tr}\left({\mathbf{a}}_i{\mathbf{b}}_i^{\mathsf{T}}{\mathbf{d}}_j{\mathbf{c}}_j^{\mathsf{T}}\right)\\ & =\sum _{i,j}\mathrm{tr}\left({\mathbf{c}}_j^{\mathsf{T}}{\mathbf{a}}_i{\mathbf{b}}_i^{\mathsf{T}}{\mathbf{d}}_j\right)\\ & =\sum _{i,j}({\mathbf{a}}_i\cdot {\mathbf{c}}_j)({\mathbf{b}}_i\cdot {\mathbf{d}}_j)\\ & =\mathbf{A}{}_{\cdot }^{\cdot }\mathbf{B}\end{array}}$

Furthermore, since,

${\displaystyle \begin{array}{cc}{\mathbf{A}}^{\mathsf{T}}& =\sum _{i,j}{\left({\mathbf{a}}_i{\mathbf{b}}_j^{\mathsf{T}}\right)}^{\mathsf{T}}\\ & =\sum _{i,j}{\mathbf{b}}_i{\mathbf{a}}_j^{\mathsf{T}}\end{array}}$

we get that,

${\displaystyle \mathbf{A}{}_{\cdot }^{\cdot }\mathbf{B}=\mathbf{A}\underset{\_}{{}_{\cdot }^{\cdot }}{\mathbf{B}}^{\mathsf{T}}}$

so the second possible definition of the double-dot product is just the first with an additional transposition on the second dyadic. For these reasons, the first definition of the double-dot product is preferred, though some authors still use the second.

Double-cross product

We can see that, for any dyad formed from two vectors a and b, its double cross product is zero.

${\displaystyle \left(\mathbf{a}\mathbf{b}\right){}_{\times }^{\times }\left(\mathbf{a}\mathbf{b}\right)=\left(\mathbf{a}\times \mathbf{a}\right)\left(\mathbf{b}\times \mathbf{b}\right)=0}$

However, by definition, a dyadic double-cross product on itself will generally be non-zero. For example, a dyadic A composed of six different vectors

${\displaystyle \mathbf{A}=\sum _{i=1}^3{\mathbf{a}}_i{\mathbf{b}}_i}$

has a non-zero self-double-cross product of

${\displaystyle \mathbf{A}{}_{\times }^{\times }\mathbf{A}=2\left[\left({\mathbf{a}}_1\times {\mathbf{a}}_2\right)\left({\mathbf{b}}_1\times {\mathbf{b}}_2\right)+\left({\mathbf{a}}_2\times {\mathbf{a}}_3\right)\left({\mathbf{b}}_2\times {\mathbf{b}}_3\right)+\left({\mathbf{a}}_3\times {\mathbf{a}}_1\right)\left({\mathbf{b}}_3\times {\mathbf{b}}_1\right)\right]}$

Tensor contraction

The spur or expansion factor arises from the formal expansion of the dyadic in a coordinate basis by replacing each dyadic product by a dot product of vectors:

${\displaystyle \begin{array}{cc}|\mathbf{A}|=& A_{11}\mathbf{i}\cdot \mathbf{i}+A_{12}\mathbf{i}\cdot \mathbf{j}+A_{13}\mathbf{i}\cdot \mathbf{k}\\ +& A_{21}\mathbf{j}\cdot \mathbf{i}+A_{22}\mathbf{j}\cdot \mathbf{j}+A_{23}\mathbf{j}\cdot \mathbf{k}\\ +& A_{31}\mathbf{k}\cdot \mathbf{i}+A_{32}\mathbf{k}\cdot \mathbf{j}+A_{33}\mathbf{k}\cdot \mathbf{k}\\ =& A_{11}+A_{22}+A_{33}\end{array}}$

in index notation this is the contraction of indices on the dyadic:

${\displaystyle |\mathbf{A}|=\sum _i{A}_i{}^i}$

In three dimensions only, the rotation factor arises by replacing every dyadic product by a cross product

${\displaystyle \begin{array}{cc}⟨\mathbf{A}⟩=& A_{11}\mathbf{i}\times \mathbf{i}+A_{12}\mathbf{i}\times \mathbf{j}+A_{13}\mathbf{i}\times \mathbf{k}\\ +& A_{21}\mathbf{j}\times \mathbf{i}+A_{22}\mathbf{j}\times \mathbf{j}+A_{23}\mathbf{j}\times \mathbf{k}\\ +& A_{31}\mathbf{k}\times \mathbf{i}+A_{32}\mathbf{k}\times \mathbf{j}+A_{33}\mathbf{k}\times \mathbf{k}\\ =& A_{12}\mathbf{k}-A_{13}\mathbf{j}-A_{21}\mathbf{k}\\ +& A_{23}\mathbf{i}+A_{31}\mathbf{j}-A_{32}\mathbf{i}\\ =& \left(A_{23}-A_{32}\right)\mathbf{i}+\left(A_{31}-A_{13}\right)\mathbf{j}+\left(A_{12}-A_{21}\right)\mathbf{k}\end{array}}$

In index notation this is the contraction of A with the Levi-Civita tensor

${\displaystyle ⟨\mathbf{A}⟩=\sum _{jk}{{ϵ}_i}^{jk}{A}_{jk}.}$

Unit dyadic

There exists a unit dyadic, denoted by I, such that, for any vector a,

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

Given a basis of 3 vectors a, b and c, with reciprocal basis ${\displaystyle \hat{\mathbf{a}},\hat{\mathbf{b}},\hat{\mathbf{c}}}$, the unit dyadic is expressed by

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

In the standard basis (for definitions of i, j, k see in the above section § Three-dimensional Euclidean space),

${\displaystyle \mathbf{I}=\mathbf{i}\mathbf{i}+\mathbf{j}\mathbf{j}+\mathbf{k}\mathbf{k}}$

Explicitly, the dot product to the right of the unit dyadic is

${\displaystyle \begin{array}{cc}\mathbf{I}\cdot \mathbf{a}& =(\mathbf{i}\mathbf{i}+\mathbf{j}\mathbf{j}+\mathbf{k}\mathbf{k})\cdot \mathbf{a}\\ & =\mathbf{i}(\mathbf{i}\cdot \mathbf{a})+\mathbf{j}(\mathbf{j}\cdot \mathbf{a})+\mathbf{k}(\mathbf{k}\cdot \mathbf{a})\\ & =\mathbf{i}{a}_x+\mathbf{j}{a}_y+\mathbf{k}{a}_z\\ & =\mathbf{a}\end{array}}$

and to the left

${\displaystyle \begin{array}{cc}\mathbf{a}\cdot \mathbf{I}& =\mathbf{a}\cdot (\mathbf{i}\mathbf{i}+\mathbf{j}\mathbf{j}+\mathbf{k}\mathbf{k})\\ & =(\mathbf{a}\cdot \mathbf{i})\mathbf{i}+(\mathbf{a}\cdot \mathbf{j})\mathbf{j}+(\mathbf{a}\cdot \mathbf{k})\mathbf{k}\\ & =a_x\mathbf{i}+a_y\mathbf{j}+a_z\mathbf{k}\\ & =\mathbf{a}\end{array}}$

The corresponding matrix is

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

This can be put on more careful foundations (explaining what the logical content of "juxtaposing notation" could possibly mean) using the language of tensor products. If V is a finite-dimensional vector space, a dyadic tensor on V is an elementary tensor in the tensor product of V with its dual space.

The tensor product of V and its dual space is isomorphic to the space of linear maps from V to V: a dyadic tensor vf is simply the linear map sending any w in V to f(w)v. When V is Euclidean n-space, we can use the inner product to identify the dual space with V itself, making a dyadic tensor an elementary tensor product of two vectors in Euclidean space.

In this sense, the unit dyadic ij is the function from 3-space to itself sending a₁i + a₂j + a₃k to a₂i, and jj sends this sum to a₂j. Now it is revealed in what (precise) sense ii + jj + kk is the identity: it sends a₁i + a₂j + a₃k to itself because its effect is to sum each unit vector in the standard basis scaled by the coefficient of the vector in that basis.

Properties of unit dyadics

${\displaystyle \begin{array}{cc}\left(\mathbf{a}\times \mathbf{I}\right)\cdot \left(\mathbf{b}\times \mathbf{I}\right)& =\mathbf{b}\mathbf{a}-\left(\mathbf{a}\cdot \mathbf{b}\right)\mathbf{I}\\ \mathbf{I}{}_{\times }^{\cdot }\left(\mathbf{a}\mathbf{b}\right)& =\mathbf{b}\times \mathbf{a}\\ \mathbf{I}{}_{\times }^{\times }\mathbf{A}& =(\mathbf{A}{}_{\cdot }^{\cdot }\mathbf{I})\mathbf{I}-{\mathbf{A}}^{\mathsf{T}}\\ \mathbf{I}{}_{\cdot }^{\cdot }\left(\mathbf{a}\mathbf{b}\right)& =\left(\mathbf{I}\cdot \mathbf{a}\right)\cdot \mathbf{b}=\mathbf{a}\cdot \mathbf{b}=\mathrm{t}\mathrm{r}\left(\mathbf{a}\mathbf{b}\right)\end{array}}$

where "tr" denotes the trace.

Examples

Vector projection and rejection

A nonzero vector a can always be split into two perpendicular components, one parallel (‖) to the direction of a unit vector n, and one perpendicular (⊥) to it;

${\displaystyle \mathbf{a}={\mathbf{a}}_{\parallel }+{\mathbf{a}}_{\perp }}$

The parallel component is found by vector projection, which is equivalent to the dot product of a with the dyadic nn,

${\displaystyle {\mathbf{a}}_{\parallel }=\mathbf{n}(\mathbf{n}\cdot \mathbf{a})=(\mathbf{n}\mathbf{n})\cdot \mathbf{a}}$

and the perpendicular component is found from vector rejection, which is equivalent to the dot product of a with the dyadic I − nn,

${\displaystyle {\mathbf{a}}_{\perp }=\mathbf{a}-\mathbf{n}(\mathbf{n}\cdot \mathbf{a})=(\mathbf{I}-\mathbf{n}\mathbf{n})\cdot \mathbf{a}}$

Rotation dyadic

2d rotations

The dyadic

${\displaystyle \mathbf{J}=\mathbf{j}\mathbf{i}-\mathbf{i}\mathbf{j}=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)}$

is a 90° anticlockwise rotation operator in 2d. It can be left-dotted with a vector r = xi + yj to produce the vector,

${\displaystyle (\mathbf{j}\mathbf{i}-\mathbf{i}\mathbf{j})\cdot (x\mathbf{i}+y\mathbf{j})=x\mathbf{j}\mathbf{i}\cdot \mathbf{i}-x\mathbf{i}\mathbf{j}\cdot \mathbf{i}+y\mathbf{j}\mathbf{i}\cdot \mathbf{j}-y\mathbf{i}\mathbf{j}\cdot \mathbf{j}=-y\mathbf{i}+x\mathbf{j},}$

in summary

${\displaystyle \mathbf{J}\cdot \mathbf{r}={\mathbf{r}}_{\mathrm{r}\mathrm{o}\mathrm{t}}}$

or in matrix notation

${\displaystyle \left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}-y\\ x\end{array}\right).}$

For any angle θ, the 2d rotation dyadic for a rotation anti-clockwise in the plane is

${\displaystyle \mathbf{R}=\mathbf{I}\cos \theta +\mathbf{J}\sin \theta =(\mathbf{i}\mathbf{i}+\mathbf{j}\mathbf{j})\cos \theta +(\mathbf{j}\mathbf{i}-\mathbf{i}\mathbf{j})\sin \theta =\left(\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right)}$

where I and J are as above, and the rotation of any 2d vector a = a_xi + a_yj is

${\displaystyle {\mathbf{a}}_{\mathrm{r}\mathrm{o}\mathrm{t}}=\mathbf{R}\cdot \mathbf{a}}$

3d rotations

A general 3d rotation of a vector a, about an axis in the direction of a unit vector ω and anticlockwise through angle θ, can be performed using Rodrigues' rotation formula in the dyadic form

${\displaystyle {\mathbf{a}}_{\mathrm{r}\mathrm{o}\mathrm{t}}=\mathbf{R}\cdot \mathbf{a},}$

where the rotation dyadic is

${\displaystyle \mathbf{R}=\mathbf{I}\cos \theta +\mathbf{\Omega }\sin \theta +\mathit{\omega }\mathit{\omega }(1-\cos \theta ),}$

and the Cartesian entries of ω also form those of the dyadic

${\displaystyle \mathbf{\Omega }={\omega }_x(\mathbf{k}\mathbf{j}-\mathbf{j}\mathbf{k})+{\omega }_y(\mathbf{i}\mathbf{k}-\mathbf{k}\mathbf{i})+{\omega }_z(\mathbf{j}\mathbf{i}-\mathbf{i}\mathbf{j}),}$

The effect of Ω on a is the cross product

${\displaystyle \mathbf{\Omega }\cdot \mathbf{a}=\mathit{\omega }\times \mathbf{a}}$

which is the dyadic form the cross product matrix with a column vector.

Lorentz transformation

In special relativity, the Lorentz boost with speed v in the direction of a unit vector n can be expressed as

${\displaystyle t^{\prime }=\gamma \left(t-\frac{v\mathbf{n}\cdot \mathbf{r}}{c^2}\right)}$

${\displaystyle {\mathbf{r}}^{\prime }=[\mathbf{I}+(\gamma -1)\mathbf{n}\mathbf{n}]\cdot \mathbf{r}-\gamma v\mathbf{n}t}$

where

${\displaystyle \gamma =\frac{1}{\sqrt{1-{\displaystyle \frac{v^2}{c^2}}}}}$

is the Lorentz factor.

Related terms

Some authors generalize from the term dyadic to related terms triadic, tetradic and polyadic.