In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with vector algebra.
There are numerous ways to multiply two Euclidean vectors.
Both of these have various significant geometric interpretations and are widely used in mathematics, physics, and engineering.
The dyadic product is distributive over vector addition, and associative with scalar multiplication.
However, the product is not commutative; changing the order of the vectors results in a different dyadic.
It also has some aspects of matrix algebra, as the numerical components of vectors can be arranged into row and column vectors, and those of second order tensors in square 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 by Josiah Willard Gibbs in 1884.
The notation and terminology are relatively obsolete today.
Its uses in physics include continuum mechanics and electromagnetism.
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.
A dyad is a tensor of order two and rank one, and is the dyadic product of two vectors (complex vectors in general), whereas a dyadic is a general tensor of order two (which may be full rank or not).
There are several equivalent terms and notations for this product: 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.
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 of the sum or equivalently an entry of the matrix) — the dyadic product of a pair of basis vectors 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: If the Euclidean space is N-dimensional, and where ei and ej are the standard basis vectors in N-dimensions (the index i on ei selects a specific vector, not a component of the vector as in ai), then in algebraic form their dyadic product is: This is known as the nonion form of the dyadic.
In this case, the forming vectors are non-coplanar,[dubious – discuss] see Chen (1983).
The following table classifies dyadics: 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.
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 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 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 index notation this is the contraction of A with the Levi-Civita tensor 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 (for definitions of i, j, k see in the above section § Three-dimensional Euclidean space), 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.
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.
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.
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; The parallel component is found by vector projection, which is equivalent to the dot product of a with the dyadic nn, and the perpendicular component is found from vector rejection, which is equivalent to the dot product of a with the dyadic I − nn, The dyadic is a 90° anticlockwise rotation operator in 2d.
It can be left-dotted with a vector r = xi + yj 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 = axi + ayj is 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 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 with a column vector.