Dyadic tensor
Encyclopedia
In multilinear algebra
, a dyadic is a second rank tensor
written in a special notation, formed by juxtaposing pairs of vectors, along with a notation for manipulating such expressions analogous to the rules for matrix algebra
.
Each component of a dyadic is a dyad. A dyad is the juxtaposition of a pair of basis vectors and a scalar coefficient. As an example, let
be a pair of three-dimensional vectors. Then the juxtaposition of A and X is
each monomial of which is a dyad. This dyadic can be represented as a 3×3 matrix
Thus a dyadic is a covariant tensor
of order two.
The dyadic itself, rather than its components, is referred to by a boldface letter A = (Aij).
:
where the vectors ei denote the coordinate basis. The resulting expression transforms like a covariant vector. This suggests employing the notation
so that the dot product associates with the juxtaposition of vectors.
The tensor contraction
of a dyadic
is the spur or expansion factor. It arises from the formal expansion of the dyadic in a coordinate basis by replacing each juxtaposition by a dot product of vectors. In three dimensions only, the rotation factor
arises by replacing every juxtaposition by a cross product
. The resulting vector is the complete contraction of A with the Levi-Civita tensor:
Multilinear algebra
In mathematics, multilinear algebra 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-vectors and multivectors with Grassmann algebra.-Origin:In a vector space...
, a dyadic is a second rank tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
written in a special notation, formed by juxtaposing pairs of vectors, along with a notation for manipulating such expressions analogous to the rules for matrix algebra
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
.
Each component of a dyadic is a dyad. A dyad is the juxtaposition of a pair of basis vectors and a scalar coefficient. As an example, let
be a pair of three-dimensional vectors. Then the juxtaposition of A and X is
each monomial of which is a dyad. This dyadic can be represented as a 3×3 matrix
Definition
Following , a dyadic (in three dimensions) is a 3×3 array of components Aij, expressed in coordinates that satisfy a covariant transformation law when passing from one coordinate system to another:Thus a dyadic is a covariant tensor
Tensor
Tensors are geometric objects that describe linear relations between vectors, scalars, and other tensors. Elementary examples include the dot product, the cross product, and linear maps. Vectors and scalars themselves are also tensors. A tensor can be represented as a multi-dimensional array of...
of order two.
The dyadic itself, rather than its components, is referred to by a boldface letter A = (Aij).
Operations on dyadics
A dyadic A can be combined with a vector v by means of the dot productDot product
In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
:
where the vectors ei denote the coordinate basis. The resulting expression transforms like a covariant vector. This suggests employing the notation
so that the dot product associates with the juxtaposition of vectors.
The tensor contraction
Tensor contraction
In multilinear algebra, a tensor contraction is an operation on one or more tensors that arises from the natural pairing of a finite-dimensional vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor caused by applying the summation...
of a dyadic
is the spur or expansion factor. It arises from the formal expansion of the dyadic in a coordinate basis by replacing each juxtaposition by a dot product of vectors. In three dimensions only, the rotation factor
arises by replacing every juxtaposition by a cross product
Cross product
In mathematics, the cross product, vector product, or Gibbs vector product is a binary operation on two vectors in three-dimensional space. It results in a vector which is perpendicular to both of the vectors being multiplied and normal to the plane containing them...
. The resulting vector is the complete contraction of A with the Levi-Civita tensor:
Examples
The dyadic tensor- J = j i − i j =
is a 90° rotation operator in two dimensions. It can be dottedDot productIn mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number obtained by multiplying corresponding entries and then summing those products...
(from the left) with a vector to produce the rotation:
or in matrix notation
A General 2-D Rotation Dyadic for
angle, anti-clockwise
The identity dyadic tensor in three dimensions is
- I = i i + j j + k k = iTi + jTj + kTk.
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 (and do) 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 dyadic tensor i j is the function from 3-space to itself sending
ai + bj + ck to bi, and j j sends this sum to bj. Now it is revealed in what (precise) sense i i + j j + k k is the identity: it sends ai + bj + ck 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.