Inmathematics, specificallymultilinear algebra, adyadic ordyadic tensor is a secondordertensor, written in a notation that fits in withvector algebra.

There are numerous ways to multiply twoEuclidean vectors. Thedot product takes in two vectors and returns ascalar, while thecross product^[a] returns apseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics,physics, andengineering. Thedyadic product takes in two vectors and returns a second order tensor called adyadic in this context. A dyadic can be used to contain physical or geometric information, although in general there is no direct way of geometrically interpreting it.

The dyadic product isdistributive overvector addition, andassociative withscalar multiplication. Therefore, the dyadic product islinear in both of its operands. In general, two dyadics can be added to get another dyadic, andmultiplied by numbers to scale the dyadic. However, the product is notcommutative; changing the order of the vectors results in a different dyadic.

The formalism ofdyadic algebra is an extension of vector algebra to include the dyadic product of vectors. The dyadic product is also associative with the dot and cross products with other vectors, which allows the dot, cross, and dyadic products to be combined to obtain other scalars, vectors, or dyadics.

It also has some aspects ofmatrix algebra, as the numerical components of vectors can be arranged intorow and column vectors, and those of second order tensors insquare matrices. Also, the dot, cross, and dyadic products can all be expressed in matrix form. Dyadic expressions may closely resemble the matrix equivalents.

The dot product of a dyadic with a vector gives another vector, and taking the dot product of this result gives a scalar derived from the dyadic. The effect that a given dyadic has on other vectors can provide indirect physical or geometric interpretations.

Dyadic notation was first established byJosiah Willard Gibbs in 1884. The notation and terminology are relatively obsolete today. Its uses in physics includecontinuum mechanics andelectromagnetism.

In this article, upper-case bold variables denote dyadics (including dyads) whereas lower-case bold variables denote vectors. An alternative notation uses respectively double and single over- or underbars.

Adyad is atensor oforder two andrank one, and is the dyadic product of twovectors (complex vectors in general), whereas adyadic is a generaltensor oforder two (which may be full rank or not).

There are several equivalent terms and notations for this product:

• thedyadic product of two vectors${\displaystyle \mathbf{a}}$ and${\displaystyle \mathbf{b}}$ is denoted by${\displaystyle \mathbf{a}\mathbf{b}}$ (juxtaposed; no symbols, multiplication signs, crosses, dots, etc.)
• theouter product of twocolumn vectors${\displaystyle \mathbf{a}}$ and${\displaystyle \mathbf{b}}$ is denoted and defined as${\displaystyle \mathbf{a}\otimes \mathbf{b}}$ or${\displaystyle \mathbf{a}{\mathbf{b}}^{\mathsf{T}}}$, where${\displaystyle \mathsf{T}}$ meanstranspose,
• thetensor product of two vectors${\displaystyle \mathbf{a}}$ and${\displaystyle \mathbf{b}}$ is denoted${\displaystyle \mathbf{a}\otimes \mathbf{b}}$,

In the dyadic context they all have the same definition and meaning, and are used synonymously, although thetensor product is an instance of the more general and abstract use of the term.

To illustrate the equivalent usage, considerthree-dimensionalEuclidean space, letting:

be two vectors wherei,j,k (also denotede₁,e₂,e₃) are the standardbasis vectors in thisvector space (see alsoCartesian coordinates). Then the dyadic product ofa andb can be represented as a sum:

or by extension from row and column vectors, a 3×3 matrix (also the result of the outer product or tensor product ofa andb):

Just as the standard basis (and unit) vectorsi,j,k, have the representations:

(which can be transposed), thestandard basis (and unit) dyads have the representation:

For a simple numerical example in the standard basis:

If the Euclidean space isN-dimensional, and

wheree_i ande_j are thestandard basis vectors inN-dimensions (the indexi one_i selects a specific vector, not a component of the vector as ina_i), then in algebraic form their dyadic product is:

${\displaystyle \mathbf{a}\mathbf{b}=\sum _{j=1}^N\sum _{i=1}^N{a}_i{b}_j{\mathbf{e}}_i{\mathbf{e}}_j.}$

This is known as thenonion form of the dyad. Their outer/tensor product in matrix form is:

Adyadic polynomialA, otherwise known as a dyadic, is formed from multiple vectorsa_i andb_j:

${\displaystyle \mathbf{A}=\sum _i{\mathbf{a}}_i{\mathbf{b}}_i={\mathbf{a}}_1{\mathbf{b}}_1+{\mathbf{a}}_2{\mathbf{b}}_2+{\mathbf{a}}_3{\mathbf{b}}_3+\dots }$

A dyadic which cannot be reduced to a sum of less thanN dyads is said to be complete. In this case, the forming vectors are non-coplanar,^{[dubious –discuss]} seeChen (1983).

The following table classifies dyadics:

	Determinant	Adjugate	Matrix and itsrank
Zero	= 0	= 0	= 0; rank 0: all zeroes
Linear	= 0	= 0	≠ 0; rank 1: at least one non-zero element and all 2 × 2 subdeterminants zero (single dyadic)
Planar	= 0	≠ 0 (single dyadic)	≠ 0; rank 2: at least one non-zero 2 × 2 subdeterminant
Complete	≠ 0	≠ 0	≠ 0; rank 3: non-zero determinant

The following identities are a direct consequence of the definition of the tensor product:^[1]

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

	Left	Right
Dot product	${\displaystyle \mathbf{c}\cdot \left(\mathbf{a}\mathbf{b}\right)=\left(\mathbf{c}\cdot \mathbf{a}\right)\mathbf{b}}$	${\displaystyle \left(\mathbf{a}\mathbf{b}\right)\cdot \mathbf{c}=\mathbf{a}\left(\mathbf{b}\cdot \mathbf{c}\right)}$
Cross product	${\displaystyle \mathbf{c}\times \left(\mathbf{a}\mathbf{b}\right)=\left(\mathbf{c}\times \mathbf{a}\right)\mathbf{b}}$	${\displaystyle \left(\mathbf{a}\mathbf{b}\right)\times \mathbf{c}=\mathbf{a}\left(\mathbf{b}\times \mathbf{c}\right)}$

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

		Dot	Cross
Dot	Dot product	Double–dot product and ${\displaystyle \mathbf{a}\mathbf{b}\underset{\_}{{}_{\cdot }^{\cdot }}\mathbf{c}\mathbf{d}=\left(\mathbf{a}\cdot \mathbf{d}\right)\left(\mathbf{b}\cdot \mathbf{c}\right)}$	Dot–cross product ${\displaystyle \left(\mathbf{a}\mathbf{b}\right){}_{\cdot }^{\times }\left(\mathbf{c}\mathbf{d}\right)=\left(\mathbf{a}\cdot \mathbf{c}\right)\left(\mathbf{b}\times \mathbf{d}\right)}$
Cross		Cross–dot product ${\displaystyle \left(\mathbf{a}\mathbf{b}\right){}_{\times }^{\cdot }\left(\mathbf{c}\mathbf{d}\right)=\left(\mathbf{a}\times \mathbf{c}\right)\left(\mathbf{b}\cdot \mathbf{d}\right)}$	Double–cross product ${\displaystyle \left(\mathbf{a}\mathbf{b}\right){}_{\times }^{\times }\left(\mathbf{c}\mathbf{d}\right)=\left(\mathbf{a}\times \mathbf{c}\right)\left(\mathbf{b}\times \mathbf{d}\right)}$

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:

		Dot	Cross
Dot	Dot product ${\displaystyle \mathbf{A}\cdot \mathbf{B}=\sum _{i,j}\left({\mathbf{b}}_i\cdot {\mathbf{c}}_j\right){\mathbf{a}}_i{\mathbf{d}}_j}$	Double–dot product and	Dot–cross product ${\displaystyle \mathbf{A}{}_{\cdot }^{\times }\mathbf{B}=\sum _{i,j}\left({\mathbf{a}}_i\cdot {\mathbf{c}}_j\right)\left({\mathbf{b}}_i\times {\mathbf{d}}_j\right)}$
Cross		Cross–dot product ${\displaystyle \mathbf{A}{}_{\times }^{\cdot }\mathbf{B}=\sum _{i,j}\left({\mathbf{a}}_i\times {\mathbf{c}}_j\right)\left({\mathbf{b}}_i\cdot {\mathbf{d}}_j\right)}$	Double–cross product ${\displaystyle \mathbf{A}{}_{\times }^{\times }\mathbf{B}=\sum _{i,j}\left({\mathbf{a}}_i\times {\mathbf{c}}_j\right)\left({\mathbf{b}}_i\times {\mathbf{d}}_j\right)}$

The first definition of the double–dot product is theFrobenius inner product,

Furthermore, since,

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.

We can see that, for any dyad formed from two vectorsa andb, 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 dyadicA 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

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

Thespur orexpansion factor arises from the formal expansion of the dyadic in a coordinate basis by replacing each dyadic product by a dot product of vectors:

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

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

In three dimensions only, therotation factor arises by replacing every dyadic product by across product

In index notation this is the contraction ofA with theLevi-Civita tensor

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

There exists a unit dyadic, denoted byI, such that, for any vectora,

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

Given a basis of 3 vectorsa,b andc, withreciprocal 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 ofi,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

and to the left

The corresponding matrix is

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. IfV is a finite-dimensionalvector space, a dyadic tensor onV is an elementary tensor in the tensor product ofV with itsdual space.

The tensor product ofV and its dual space isisomorphic to the space oflinear maps fromV toV: a dyadic tensorvf is simply the linear map sending anyw inV tof(w)v. WhenV is Euclideann-space, we can use theinner product to identify the dual space withV itself, making a dyadic tensor an elementary tensor product of two vectors in Euclidean space.

In this sense, the unit dyadicij is the function from 3-space to itself sendinga₁i +a₂j +a₃k toa₂i, andjj sends this sum toa₂j. Now it is revealed in what (precise) senseii +jj +kk is the identity: it sendsa₁i +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.

where "tr" denotes thetrace.

A nonzero vectora can always be split into two perpendicular components, one parallel (‖) to the direction of aunit vectorn, and one perpendicular (⊥) to it;

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

The parallel component is found byvector projection, which is equivalent to the dot product ofa with the dyadicnn,

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

and the perpendicular component is found fromvector rejection, which is equivalent to the dot product ofa with the dyadicI −nn,

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

The dyadic

is a 90° anticlockwiserotation operator in 2d. It can be left-dotted with a vectorr =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

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

whereI andJ are as above, and the rotation of any 2d vectora =a_xi +a_yj is

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

A general 3d rotation of a vectora, about an axis in the direction of aunit vectorω and anticlockwise through angleθ, can be performed usingRodrigues' 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 ofω 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 ofΩ ona is the cross product

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

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

Inspecial relativity, theLorentz boost with speedv in the direction of a unit vectorn 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 theLorentz factor.

Some authors generalize from the termdyadic to related termstriadic,tetradic andpolyadic.^[2]

• Kronecker product
• Bivector
• Polyadic algebra
• Unit vector
• Multivector
• Differential form
• Quaternions
• Field (mathematics)

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• Vector Analysis, a Text-Book for the use of Students of Mathematics and Physics, Founded upon the Lectures of J. Willard Gibbs PhD LLD, Edwind Bidwell Wilson PhD
• Advanced Field Theory, I.V.Lindel
• Vector and Dyadic Analysis
• Introductory Tensor Analysis
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• Nasa.gov, An introduction to Tensors for students of Physics and Engineering, J.C. Kolecki

• Tensors

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