TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, specifically
multilinear algebra In mathematics, multilinear algebra extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector space, vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of Multivect ...
order Order, ORDER or Orders may refer to: * Orderliness Orderliness is a quality that is characterized by a person’s interest in keeping their surroundings and themselves well organized, and is associated with other qualities such as cleanliness a ...
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

, written in a notation that fits in with
vector algebra In mathematics, vector algebra may mean: * Linear algebra Linear algebra is the branch of mathematics concerning linear equations such as: :a_1x_1+\cdots +a_nx_n=b, linear maps such as: :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their ...
. There are numerous ways to multiply two
Euclidean vector In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
s. The
dot product In mathematics, the dot product or scalar productThe term ''scalar product'' is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space. is an algebraic operation that takes two equal-length seque ...
takes in two vectors and returns a
scalar Scalar may refer to: *Scalar (mathematics), an element of a field, which is used to define a vector space, usually the field of real numbers *Scalar (physics), a physical quantity that can be described by a single element of a number field such as ...
, while the
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

returns a
pseudovector In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function (mathematics), function of some vector (geometry), vectors or other geometric shapes, that resembles a vector, and behaves like a vector in ma ...

. Both of these have various significant geometric interpretations and are widely used in mathematics,
physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ...

, and
engineering Engineering is the use of scientific principles to design and build machines, structures, and other items, including bridges, tunnels, roads, vehicles, and buildings. The discipline of engineering encompasses a broad range of more speciali ...

. The dyadic product takes in two vectors and returns a second order tensor called a ''dyadic'' 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 is distributive over
vector addition Vector may refer to: Biology *Vector (epidemiology) In epidemiology, a disease vector is any living agent that carries and transmits an infectious pathogen to another living organism; agents regarded as vectors are organisms, such as Para ...

, and
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

# Definitions and terminology

## Dyadic, outer, and tensor products

tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of Tensor order, order two and Tensor rank, rank one, and is the dyadic product of two Euclidean vector, vectors (complex vectors in general), whereas a ''dyadic'' is a general
tensor In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of Tensor order, order two (which may be full rank or not). There are several equivalent terms and notations for this product: * the dyadic product of two vectors $\mathbf$ and $\mathbf$ is denoted by $\mathbf\mathbf$ (juxtaposed; no symbols, multiplication signs, crosses, dots, etc.) * the outer product of two column vectors $\mathbf$ and $\mathbf$ is denoted and defined as $\mathbf \otimes \mathbf$ or $\mathbf\mathbf^\mathsf$, where $\mathsf$ means transpose, * the tensor product of two vectors $\mathbf$ and $\mathbf$ is denoted $\mathbf \otimes \mathbf$, In the dyadic context they all have the same definition and meaning, and are used synonymously, although the tensor product is an instance of the more general and abstract use of the term. Dirac's bra–ket notation makes the use of dyads and dyadics intuitively clear, see #Cahill, Cahill (2013).

### Three-dimensional Euclidean space

To illustrate the equivalent usage, consider Three-dimensional space, three-dimensional Euclidean space, letting: :$\begin \mathbf &= a_1 \mathbf + a_2 \mathbf + a_3 \mathbf \\ \mathbf &= b_1 \mathbf + b_2 \mathbf + b_3 \mathbf \end$ be two vectors where i, j, k (also denoted e1, e2, e3) are the standard basis vectors in this vector space (see also Cartesian coordinates). Then the dyadic product of a and b can be represented as a sum: :$\begin \mathbf =\qquad &a_1 b_1 \mathbf + a_1 b_2 \mathbf + a_1 b_3 \mathbf \\ + &a_2 b_1 \mathbf + a_2 b_2 \mathbf + a_2 b_3 \mathbf \\ + &a_3 b_1 \mathbf + a_3 b_2 \mathbf + a_3 b_3 \mathbf \end$ or by extension from row and column vectors, a 3×3 matrix (also the result of the outer product or tensor product of a and b): :$\mathbf \equiv \mathbf\otimes\mathbf \equiv \mathbf^\mathsf = \begin a_1 \\ a_2 \\ a_3 \end\begin b_1 & b_2 & b_3 \end = \begin a_1 b_1 & a_1 b_2 & a_1 b_3 \\ a_2 b_1 & a_2 b_2 & a_2 b_3 \\ a_3 b_1 & a_3 b_2 & a_3 b_3 \end.$ A ''dyad'' is a component of the dyadic (a monomial of the sum or equivalently an entry of the matrix) — the dyadic product of a pair of basis vectors scalar multiplication, scalar multiplied by a number. Just as the standard basis (and unit) vectors i, j, k, have the representations: :$\begin \mathbf &= \begin 1 \\ 0 \\ 0 \end,& \mathbf &= \begin 0 \\ 1 \\ 0 \end,& \mathbf &= \begin 0 \\ 0 \\ 1 \end \end$ (which can be transposed), the ''standard basis (and unit) dyads'' have the representation: :$\begin \mathbf &= \begin 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end, & \mathbf &= \begin 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end, & \mathbf &= \begin 0 & 0 & 1 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end \\ \mathbf &= \begin 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end, & \mathbf &= \begin 0 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end, & \mathbf &= \begin 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end \\ \mathbf &= \begin 0 & 0 & 0 \\ 0 & 0 & 0 \\ 1 & 0 & 0 \end, & \mathbf &= \begin 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0 \end, & \mathbf &= \begin 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end \end$ For a simple numerical example in the standard basis: :$\begin \mathbf &= 2\mathbf + \frac\mathbf - 8\pi\mathbf + \frac\mathbf \\\left[2pt\right] &= 2 \begin 0 & 1 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end + \frac\begin 0 & 0 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 0 \end - 8\pi \begin 0 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end + \frac\begin 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 1 \end \\\left[2pt\right] &= \begin 0 & 2 & 0 \\ \frac & 0 & -8\pi \\ 0 & 0 & \frac \end \end$

### ''N''-dimensional Euclidean space

If the Euclidean space is ''N''-dimensional, and :$\begin \mathbf &= \sum_^N a_i\mathbf_i = a_1 \mathbf_1 + a_2 \mathbf_2 + + a_N \mathbf_N \\ \mathbf &= \sum_^N b_j\mathbf_j = b_1 \mathbf_1 + b_2 \mathbf_2 + \ldots + b_N \mathbf_N \end$ where e''i'' and e''j'' are the standard basis vectors in ''N''-dimensions (the index ''i'' on e''i'' selects a specific vector, not a component of the vector as in ''ai''), then in algebraic form their dyadic product is: :$\mathbf = \sum_^N \sum_^N a_i b_j \mathbf_i \mathbf_j.$ This is known as the ''nonion form'' of the dyadic. Their outer/tensor product in matrix form is: :$\mathbf = \mathbf^\mathsf = \begin a_1 \\ a_2 \\ \vdots \\ a_N \end\begin b_1 & b_2 & \cdots & b_N \end = \begin a_1 b_1 & a_1 b_2 & \cdots & a_1 b_N \\ a_2 b_1 & a_2 b_2 & \cdots & a_2 b_N \\ \vdots & \vdots & \ddots & \vdots \\ a_N b_1 & a_N b_2 & \cdots & a_N b_N \end.$ A ''dyadic polynomial'' A, otherwise known as a dyadic, is formed from multiple vectors a''i'' and b''j'': :$\mathbf = \sum_i\mathbf_i\mathbf_i = \mathbf_1\mathbf_1 + \mathbf_2\mathbf_2 + \mathbf_3\mathbf_3 + \ldots$ A dyadic which cannot be reduced to a sum of less than ''N'' dyads is said to be complete. In this case, the forming vectors are non-coplanar, see #Chen, Chen (1983).

## Classification

The following table classifies dyadics: :

## Identities

The following identities are a direct consequence of the definition of the tensor product:

## Product of dyadic and vector

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

There are five operations for a dyadic to another dyadic. Let a, b, c, d be real vectors. Then: : Letting :$\mathbf = \sum_i \mathbf_i\mathbf_i,\quad \mathbf = \sum_j \mathbf_j\mathbf_j$ be two general dyadics, we have: :

### Double-dot product

The first definition of the double-dot product is the Frobenius inner product, :$\begin \operatorname\left\left(\mathbf\mathbf^\mathsf\right\right) &=\sum_ \operatorname\left\left(\mathbf_i \mathbf_i^\mathsf \mathbf_j \mathbf_j^\mathsf\right\right) \\ &=\sum_ \operatorname\left\left(\mathbf_j^\mathsf \mathbf_i \mathbf_i^\mathsf \mathbf_j\right\right) \\ &=\sum_ \left(\mathbf_i\cdot\mathbf_j\right)\left(\mathbf_i\cdot\mathbf_j\right) \\ &=\mathbf _\centerdot^\centerdot \mathbf \end$ Furthermore, since, :$\begin \mathbf^\mathsf &=\sum_ \left\left(\mathbf_i\mathbf_j^\mathsf\right\right)^\mathsf \\ &=\sum_ \mathbf_i\mathbf_j^\mathsf \end$ we get that, :$\mathbf _\centerdot^\centerdot \mathbf = \mathbf \underline \mathbf^\mathsf$ 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. :$\left\left(\mathbf\right\right) _\times^\times \left\left(\mathbf\right\right) = \left\left(\mathbf\times\mathbf\right\right)\left\left(\mathbf\times\mathbf\right\right) = 0$ 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 :$\mathbf = \sum_^3 \mathbf_i\mathbf_i$ has a non-zero self-double-cross product of :$\mathbf _\times^\times \mathbf = 2\left\left[ \left\left(\mathbf_1 \times \mathbf_2\right\right)\left\left(\mathbf_1 \times \mathbf_2\right\right) + \left\left(\mathbf_2 \times \mathbf_3\right\right)\left\left(\mathbf_2 \times \mathbf_3\right\right) + \left\left(\mathbf_3 \times \mathbf_1\right\right)\left\left(\mathbf_3 \times \mathbf_1\right\right) \right\right]$

### 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: :$\begin , \mathbf, =\qquad &A_ \mathbf\cdot\mathbf + A_ \mathbf\cdot\mathbf + A_ \mathbf\cdot\mathbf \\ + &A_ \mathbf\cdot\mathbf + A_ \mathbf\cdot\mathbf + A_ \mathbf\cdot\mathbf \\ + &A_ \mathbf\cdot\mathbf + A_ \mathbf\cdot\mathbf + A_ \mathbf\cdot\mathbf \\\left[6pt\right] =\qquad &A_ + A_ + A_ \end$ in index notation this is the contraction of indices on the dyadic: :$, \mathbf, = \sum_i A_i^i$ In three dimensions only, the ''rotation factor'' arises by replacing every dyadic product by a
cross product In , the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a on two s in a three-dimensional (named here E), and is denoted by the symbol \times. Given two and , the cross produc ...

:$\begin \langle\mathbf\rangle =\qquad &A_ \mathbf\times\mathbf + A_ \mathbf\times\mathbf + A_ \mathbf\times\mathbf \\ + &A_ \mathbf\times\mathbf + A_ \mathbf\times\mathbf + A_ \mathbf\times\mathbf\\ + &A_ \mathbf\times\mathbf + A_ \mathbf\times\mathbf + A_ \mathbf\times\mathbf \\\left[6pt\right] =\qquad &A_ \mathbf - A_ \mathbf - A_ \mathbf \\ + &A_ \mathbf + A_ \mathbf - A_ \mathbf \\\left[6pt\right] =\qquad &\left\left(A_ - A_\right\right)\mathbf + \left\left(A_ - A_\right\right)\mathbf + \left\left(A_ - A_\right\right)\mathbf\\ \end$ In index notation this is the contraction of A with the Levi-Civita tensor :$\langle\mathbf\rangle = \sum_^A_.$

There exists a unit dyadic, denoted by I, such that, for any vector a, :$\mathbf\cdot\mathbf=\mathbf\cdot\mathbf= \mathbf$ Given a basis of 3 vectors a, b and c, with Dual basis, reciprocal basis $\hat, \hat, \hat$, the unit dyadic is expressed by :$\mathbf = \mathbf\hat + \mathbf\hat + \mathbf\hat$ In the standard basis, :$\mathbf = \mathbf + \mathbf + \mathbf$ Explicitly, the dot product to the right of the unit dyadic is :$\begin \mathbf \cdot \mathbf & = \left(\mathbf\mathbf + \mathbf\mathbf + \mathbf\mathbf\right)\cdot \mathbf \\ & = \mathbf\left(\mathbf \cdot \mathbf\right) + \mathbf\left(\mathbf \cdot \mathbf\right) + \mathbf \left(\mathbf \cdot \mathbf\right) \\ & = \mathbf a_x + \mathbf a_y + \mathbf a_z \\ & = \mathbf \end$ and to the left :$\begin \mathbf \cdot \mathbf & = \mathbf \cdot \left(\mathbf\mathbf + \mathbf\mathbf + \mathbf\mathbf\right)\\ & = \left(\mathbf\cdot \mathbf\right)\mathbf + \left(\mathbf\cdot \mathbf\right)\mathbf + \left(\mathbf\cdot \mathbf\right)\mathbf \\ & = a_x \mathbf + a_y \mathbf + a_z \mathbf \\ & = \mathbf \end$ The corresponding matrix is :$\mathbf=\begin 1 & 0 & 0\\ 0 & 1 & 0\\ 0 & 0 & 1\\ \end$ 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 ''a''1i + ''a''2j + ''a''3k to ''a''2i, and jj sends this sum to ''a''2j. Now it is revealed in what (precise) sense ii + jj + kk is the identity: it sends ''a''1i + ''a''2j + ''a''3k 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.

:$\begin \left\left(\mathbf\times\mathbf\right\right)\cdot\left\left(\mathbf\times\mathbf\right\right) &= \mathbf - \left\left(\mathbf\cdot\mathbf\right\right)\mathbf \\ \mathbf _\times^ \left\left(\mathbf\right\right) &= \mathbf\times\mathbf \\ \mathbf _\times^\times \mathbf &= \left(\mathbf _^ \mathbf\right)\mathbf - \mathbf^\mathsf \\ \mathbf _^ \left\left(\mathbf\right\right) &= \left\left(\mathbf\cdot\mathbf\right\right)\cdot\mathbf = \mathbf\cdot\mathbf = \mathrm\left\left(\mathbf\right\right) \end$ where "tr" denotes the Trace (linear algebra), 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; :$\mathbf = \mathbf_\parallel + \mathbf_\perp$ The parallel component is found by vector projection, which is equivalent to the dot product of a with the dyadic nn, :$\mathbf_\parallel = \mathbf\left(\mathbf\cdot\mathbf\right) = \left(\mathbf\right)\cdot\mathbf$ and the perpendicular component is found from vector rejection, which is equivalent to the dot product of a with the dyadic , :$\mathbf_\perp = \mathbf - \mathbf\left(\mathbf\cdot\mathbf\right) = \left(\mathbf - \mathbf\right)\cdot\mathbf$

### 2d rotations

The dyadic :$\mathbf = \mathbf - \mathbf = \begin 0 & -1 \\ 1 & 0 \end$ is a 90° anticlockwise Rotation operator (vector space), rotation operator in 2d. It can be left-dotted with a vector r = ''x''i + ''y''j to produce the vector, :$\left(\mathbf - \mathbf\right) \cdot \left(x \mathbf + y \mathbf\right) = x \mathbf \cdot \mathbf - x \mathbf \cdot \mathbf + y \mathbf \cdot \mathbf - y \mathbf \cdot \mathbf = -y \mathbf + x \mathbf,$ in summary :$\mathbf\cdot\mathbf = \mathbf_\mathrm$ or in matrix notation :$\begin 0 & -1 \\ 1 & 0 \end \begin x \\ y \end= \begin -y \\ x \end.$ For any angle ''θ'', the 2d rotation dyadic for a rotation anti-clockwise in the plane is :$\mathbf = \mathbf\cos\theta + \mathbf\sin\theta = \left(\mathbf+\mathbf\right)\cos\theta + \left(\mathbf-\mathbf\right)\sin\theta = \begin \cos\theta &-\sin\theta \\ \sin\theta &\;\cos\theta \end$ where I and J are as above, and the rotation of any 2d vector a = ''ax''i + ''ay''j is :$\mathbf_\mathrm = \mathbf\cdot\mathbf$

### 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 :$\mathbf_\mathrm = \mathbf \cdot \mathbf \,,$ where the rotation dyadic is :$\mathbf = \mathbf \cos\theta + \sin\theta \boldsymbol + \left(1 - \cos\theta\right) \boldsymbol \,,$ and the Cartesian entries of ω also form those of the dyadic :$\boldsymbol = \omega_x\left( \mathbf -\mathbf \right) + \omega_y\left( \mathbf -\mathbf \right) + \omega_z\left( \mathbf - \mathbf \right) \,,$ The effect of Ω on a is the cross product :$\boldsymbol\cdot\mathbf = \boldsymbol\times \mathbf$ 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 :$t\text{'} = \gamma\left\left(t - \frac \right\right)$ :$\mathbf\text{'} = \left[\mathbf + \left(\gamma-1\right) \mathbf\right]\cdot \mathbf - \gamma v \mathbft$ where :$\gamma=\frac$ is the Lorentz factor.

# Related terms

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

# References

* Chapter 2 * * . * . * . * . *

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