In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
multilinear algebra
Multilinear algebra is a subfield of mathematics that extends the methods of linear algebra. Just as linear algebra is built on the concept of a vector and develops the theory of vector spaces, multilinear algebra builds on the concepts of ''p' ...
, a dyadic or dyadic tensor is a second
order tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
, written in a notation that fits in with
vector algebra In mathematics, vector algebra may mean:
* Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space.
* The algebraic operations in vector calculus, namely the specific additional stru ...
.
There are numerous ways to multiply two
Euclidean vector
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...
s. The
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
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 ...
, while the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
returns a
pseudovector
In physics and mathematics, a pseudovector (or axial vector) is a quantity that is defined as a function of some vectors or other geometric shapes, that resembles a vector, and behaves like a vector in many situations, but is changed into its o ...
. Both of these have various significant geometric interpretations and are widely used in mathematics,
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, and
engineering
Engineering is the use of scientific method, 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 rang ...
. 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
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors ac ...
, and
associative with
scalar multiplication
In mathematics, scalar multiplication is one of the basic operations defining a vector space in linear algebra (or more generally, a module in abstract algebra). In common geometrical contexts, scalar multiplication of a real Euclidean vector by ...
. Therefore, the dyadic product is
linear
Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...
in both of its operands. In general, two dyadics can be added to get another dyadic, and
multiplied by numbers to scale the dyadic. However, the product is not
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name o ...
; changing the order of the vectors results in a different dyadic.
The formalism of dyadic 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 of
matrix algebra
In abstract algebra, a matrix ring is a set of matrices with entries in a ring ''R'' that form a ring under matrix addition and matrix multiplication . The set of all matrices with entries in ''R'' is a matrix ring denoted M''n''(''R'')Lang, ''U ...
, as the numerical components of vectors can be arranged into
row and column vectors
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
, and those of second order tensors in
square matrices
In mathematics, a square matrix is a matrix with the same number of rows and columns. An ''n''-by-''n'' matrix is known as a square matrix of order Any two square matrices of the same order can be added and multiplied.
Square matrices are often ...
. 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 by
Josiah Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
in 1884. The notation and terminology are relatively obsolete today. Its uses in physics include
continuum mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such m ...
and
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
.
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.
Definitions and terminology
Dyadic, outer, and tensor products
A ''dyad'' is a
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
of
order two and
rank
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking, such as:
Level or position in a hierarchical organization
* Academic rank
* Diplomatic rank
* Hierarchy
* ...
one, and is the dyadic product of two
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
s (
complex vector
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but ca ...
s in general), whereas a ''dyadic'' is a general
tensor
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects related to a vector space. Tensors may map between different objects such as vectors, scalars, and even other tenso ...
of
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
and
is denoted by
(juxtaposed; no symbols, multiplication signs, crosses, dots, etc.)
* the
outer product of two
column vector
In linear algebra, a column vector with m elements is an m \times 1 matrix consisting of a single column of m entries, for example,
\boldsymbol = \begin x_1 \\ x_2 \\ \vdots \\ x_m \end.
Similarly, a row vector is a 1 \times n matrix for some n, c ...
s
and
is denoted and defined as
or
, where
means
transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal;
that is, it switches the row and column indices of the matrix by producing another matrix, often denoted by (among other notations).
The tr ...
,
* the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W ...
of two vectors
and
is denoted
,
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 (2013).
Three-dimensional Euclidean space
To illustrate the equivalent usage, consider
three-dimensional
Three-dimensional space (also: 3D space, 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called ''parameters'') are required to determine the position of an element (i.e., point). This is the informal ...
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
, letting:
:
be two vectors where i, j, k (also denoted e
1, e
2, e
3) are the standard
basis vectors
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as componen ...
in this
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
(see also
Cartesian coordinates). Then the dyadic product of a and b 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 of a and b):
:
A ''dyad'' is a component of the dyadic (a
monomial
In mathematics, a monomial is, roughly speaking, a polynomial which has only one term. Two definitions of a monomial may be encountered:
# A monomial, also called power product, is a product of powers of variables with nonnegative integer expone ...
of the sum or equivalently an entry of the matrix) — the dyadic product of a pair of
basis vector
In mathematics, a set of vectors in a vector space is called a basis if every element of may be written in a unique way as a finite linear combination of elements of . The coefficients of this linear combination are referred to as components ...
s
scalar multiplied by a number.
Just as the standard basis (and unit) vectors i, j, k, have the representations:
:
(which can be transposed), the ''standard basis (and unit) dyads'' have the representation:
:
For a simple numerical example in the standard basis:
:
''N''-dimensional Euclidean space
If the Euclidean space is ''N''-
dimension
In physics and mathematics, the dimension of a Space (mathematics), mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any Point (geometry), point within it. Thus, a Line (geometry), lin ...
al, and
:
where e
''i'' and e
''j'' are the
standard basis
In mathematics, the standard basis (also called natural basis or canonical basis) of a coordinate vector space (such as \mathbb^n or \mathbb^n) is the set of vectors whose components are all zero, except one that equals 1. For example, in the c ...
vectors in ''N''-dimensions (the index ''i'' on e
''i'' selects a specific vector, not a component of the vector as in ''a
i''), then in algebraic form their dyadic product is:
:
This is known as the ''nonion form'' of the dyadic. Their outer/tensor product in matrix form is:
:
A ''dyadic polynomial'' A, otherwise known as a dyadic, is formed from multiple vectors a
''i'' and b
''j'':
:
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 (1983).
Classification
The following table classifies dyadics:
:
Identities
The following identities are a direct consequence of the definition of the tensor product:
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
:
be two general dyadics, we have:
:
Double-dot product
The first definition of the double-dot product is the
Frobenius inner product
In mathematics, the Frobenius inner product is a binary operation that takes two matrices and returns a scalar. It is often denoted \langle \mathbf,\mathbf \rangle_\mathrm. The operation is a component-wise inner product of two matrices as though ...
,
:
Furthermore, since,
:
we get that,
:
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.
:
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
:
has a non-zero self-double-cross product of
:
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:
:
in index notation this is the contraction of indices on the dyadic:
:
In three dimensions only, the ''rotation factor'' arises by replacing every dyadic product by a
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
:
In index notation this is the contraction of A with the
Levi-Civita tensor
In mathematics, particularly in linear algebra, tensor analysis, and differential geometry, the Levi-Civita symbol or Levi-Civita epsilon represents a collection of numbers; defined from the sign of a permutation of the natural numbers , for s ...
:
Unit dyadic
There exists a unit dyadic, denoted by I, such that, for any vector a,
:
Given a basis of 3 vectors a, b and c, with
reciprocal basis , the unit dyadic is expressed by
:
In the standard basis,
:
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. If ''V'' is a finite-dimensional
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called ''vectors'', may be added together and multiplied ("scaled") by numbers called '' scalars''. Scalars are often real numbers, but can ...
, a dyadic tensor on ''V'' is an elementary tensor in the tensor product of ''V'' with its
dual space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
.
The tensor product of ''V'' and its dual space is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
to the space of
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a Map (mathematics), mapping V \to W between two vect ...
s 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
In mathematics, an inner product space (or, rarely, a Hausdorff pre-Hilbert space) is a real vector space or a complex vector space with an operation called an inner product. The inner product of two vectors in the space is a scalar, often ...
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.
Properties of unit dyadics
:
where "tr" denotes the
trace
Trace may refer to:
Arts and entertainment Music
* ''Trace'' (Son Volt album), 1995
* ''Trace'' (Died Pretty album), 1993
* Trace (band), a Dutch progressive rock band
* ''The Trace'' (album)
Other uses in arts and entertainment
* ''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
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
n, and one perpendicular (⊥) to it;
:
The parallel component is found by
vector projection
The vector projection of a vector on (or onto) a nonzero vector , sometimes denoted \operatorname_\mathbf \mathbf (also known as the vector component or vector resolution of in the direction of ), is the orthogonal projection of onto a straig ...
, which is equivalent to the dot product of a with the dyadic nn,
:
and the perpendicular component is found from
vector rejection
The vector projection of a vector on (or onto) a nonzero vector , sometimes denoted \operatorname_\mathbf \mathbf (also known as the vector component or vector resolution of in the direction of ), is the orthogonal projection of onto a straig ...
, which is equivalent to the dot product of a with the dyadic ,
:
Rotation dyadic
2d rotations
The dyadic
:
is a 90° anticlockwise
rotation operator in 2d. It can be left-dotted with a vector r = ''x''i + ''y''j to produce the vector,
:
in summary
:
or in matrix notation
:
For any angle ''θ'', the 2d rotation dyadic for a rotation anti-clockwise in the plane is
:
where I and J are as above, and the rotation of any 2d vector a = ''a
x''i + ''a
y''j is
:
3d rotations
A general 3d rotation of a vector a, about an axis in the direction of a
unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
ω and anticlockwise through angle ''θ'', can be performed using
Rodrigues' rotation formula
In the theory of three-dimensional rotation, Rodrigues' rotation formula, named after Olinde Rodrigues, is an efficient algorithm for rotating a vector in space, given an axis and angle of rotation. By extension, this can be used to transform al ...
in the dyadic form
:
where the rotation dyadic is
:
and the Cartesian entries of ω also form those of the dyadic
:
The effect of Ω on a is the cross product
:
which is the dyadic form the
cross product matrix
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and i ...
with a column vector.
Lorentz transformation
In
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
, the
Lorentz boost
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
with speed ''v'' in the direction of a unit vector n can be expressed as
:
:
where
:
is the
Lorentz factor
The Lorentz factor or Lorentz term is a quantity expressing how much the measurements of time, length, and other physical properties change for an object while that object is moving. The expression appears in several equations in special relativit ...
.
Related terms
Some authors generalize from the term ''dyadic'' to related terms ''triadic'', ''tetradic'' and ''polyadic''.
[For example, ]
See also
*
Kronecker product
In mathematics, the Kronecker product, sometimes denoted by ⊗, is an operation on two matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer product (which is denoted by the same symbol) from vectors ...
*
Bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
*
Polyadic algebra Polyadic algebras (more recently called Halmos algebras) are algebraic structures introduced by Paul Halmos. They are related to first-order logic analogous to the relationship between Boolean algebras and propositional logic (see Lindenbaum–Tar ...
*
Unit vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat").
The term ''direction vecto ...
*
Multivector
*
Differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
*
Quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quater ...
*
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which i ...
Notes
References
* Chapter 2
*
* .
* .
* .
*
.
*
External links
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 PhDAdvanced Field Theory, I.V.LindelVector and Dyadic AnalysisIntroductory Tensor AnalysisNasa.gov, Foundations of Tensor Analysis for students of Physics and Engineering with an Introduction to the Theory of Relativity, J.C. KoleckiNasa.gov, An introduction to Tensors for students of Physics and Engineering, J.C. Kolecki
{{Tensors
Tensors