Dyadics

 ( Tensors ) 

In mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second order tensor, written in a notation that fits in with Vector calculus.

There are numerous ways to multiply two . The dot product takes in two vectors and returns a scalar, while the cross product returns a pseudovector. Both of these have various significant geometric interpretations and are widely used in mathematics, physics, and engineering. 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, and associative with scalar multiplication. Therefore, the dyadic product is linear 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; 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, as the numerical components of vectors can be arranged into row and column vectors, and those of second order tensors in square matrix. 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.

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Definitions and terminology

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Dyadic, outer, and tensor products A dyad is a tensor of Tensor order two and Tensor rank one, and is the dyadic product of two Euclidean vector ( in general), whereas a dyadic is a general tensor of Tensor 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 $\backslash mathbf\{a\}$ and $\backslash mathbf\{b\}$ is denoted by $\backslash mathbf\{a\}\backslash mathbf\{b\}$ (juxtaposed; no symbols, multiplication signs, crosses, dots, etc.)
• the outer product of two $\backslash mathbf\{a\}$ and $\backslash mathbf\{b\}$ is denoted and defined as $\backslash mathbf\{a\}\; \backslash otimes\; \backslash mathbf\{b\}$ or $\backslash mathbf\{a\}\backslash mathbf\{b\}^\backslash mathsf\{T\}$, where $\backslash mathsf\{T\}$ means transpose,
• the tensor product of two vectors $\backslash mathbf\{a\}$ and $\backslash mathbf\{b\}$ is denoted $\backslash mathbf\{a\}\; \backslash otimes\; \backslash mathbf\{b\}$,

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.

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Three-dimensional Euclidean space To illustrate the equivalent usage, consider three-dimensional Euclidean space, letting:

$\backslash begin\{align\}$

 \mathbf{a} &= a_1 \mathbf{i} + a_2 \mathbf{j} + a_3 \mathbf{k} \\
 \mathbf{b} &= b_1 \mathbf{i} + b_2 \mathbf{j} + b_3 \mathbf{k}
     

\end{align} 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:

$\backslash begin\{align\}$

 \mathbf{ab} =\qquad
         &a_1 b_1 \mathbf{ii} + a_1 b_2 \mathbf{ij} + a_1 b_3 \mathbf{ik} \\
   {}+{} &a_2 b_1 \mathbf{ji} + a_2 b_2 \mathbf{jj} + a_2 b_3 \mathbf{jk} \\
   {}+{} &a_3 b_1 \mathbf{ki} + a_3 b_2 \mathbf{kj} + a_3 b_3 \mathbf{kk}
     

\end{align} 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{a b} \equiv
 \mathbf{a}\otimes\mathbf{b} \equiv \mathbf{a b}^\mathsf{T} =
 \begin{pmatrix}
   a_1 \\ a_2 \\ a_3
 \end{pmatrix}\begin{pmatrix}
   b_1 & b_2 & b_3
 \end{pmatrix} = \begin{pmatrix}
   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{pmatrix}.
     

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 scalar multiplied by a number.

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

$\backslash begin\{align\}$

 \mathbf{i} &= \begin{pmatrix}
   1 \\ 0 \\ 0
 \end{pmatrix},&
 \mathbf{j} &= \begin{pmatrix}
   0 \\ 1 \\ 0
 \end{pmatrix},&
 \mathbf{k} &= \begin{pmatrix}
   0 \\ 0 \\ 1
 \end{pmatrix}
     

\end{align} (which can be transposed), the standard basis (and unit) dyads have the representation:

$\backslash begin\{align\}$

 \mathbf{ii} &= \begin{pmatrix}
   1 & 0 & 0 \\
   0 & 0 & 0 \\
   0 & 0 & 0
 \end{pmatrix}, &
 \mathbf{ij} &= \begin{pmatrix}
   0 & 1 & 0 \\
   0 & 0 & 0 \\
   0 & 0 & 0
 \end{pmatrix}, &
 \mathbf{ik} &= \begin{pmatrix}
   0 & 0 & 1 \\
   0 & 0 & 0 \\
   0 & 0 & 0
 \end{pmatrix} \\
 \mathbf{ji} &= \begin{pmatrix}
   0 & 0 & 0 \\
   1 & 0 & 0 \\
   0 & 0 & 0
 \end{pmatrix}, &
 \mathbf{jj} &= \begin{pmatrix}
   0 & 0 & 0 \\
   0 & 1 & 0 \\
   0 & 0 & 0
 \end{pmatrix}, &
 \mathbf{jk} &= \begin{pmatrix}
   0 & 0 & 0 \\
   0 & 0 & 1 \\
   0 & 0 & 0
 \end{pmatrix} \\
 \mathbf{ki} &= \begin{pmatrix}
   0 & 0 & 0 \\
   0 & 0 & 0 \\
   1 & 0 & 0
 \end{pmatrix}, &
 \mathbf{kj} &= \begin{pmatrix}
   0 & 0 & 0 \\
   0 & 0 & 0 \\
   0 & 1 & 0
 \end{pmatrix}, &
 \mathbf{kk} &= \begin{pmatrix}
   0 & 0 & 0 \\
   0 & 0 & 0 \\
   0 & 0 & 1
 \end{pmatrix}
     

\end{align}

For a simple numerical example in the standard basis:

$\backslash begin\{align\}$

 \mathbf{A} &= 2\mathbf{ij} + \frac{\sqrt{3}}{2}\mathbf{ji} - 8\pi\mathbf{jk} + \frac{2\sqrt{2}}{3}\mathbf{kk} \\[2pt]
            &= 2 \begin{pmatrix}
                   0 & 1 & 0 \\
                   0 & 0 & 0 \\
                   0 & 0 & 0
                 \end{pmatrix} + \frac{\sqrt{3}}{2}\begin{pmatrix}
                   0 & 0 & 0 \\
                   1 & 0 & 0 \\
                   0 & 0 & 0
                 \end{pmatrix} - 8\pi \begin{pmatrix}
                   0 & 0 & 0 \\
                   0 & 0 & 1 \\
                   0 & 0 & 0
                 \end{pmatrix} + \frac{2\sqrt{2}}{3}\begin{pmatrix}
                   0 & 0 & 0 \\
                   0 & 0 & 0 \\
                   0 & 0 & 1
                 \end{pmatrix} \\[2pt]
            &= \begin{pmatrix}
                                  0 & 2 &                   0 \\
                 \frac{\sqrt{3}}{2} & 0 &               -8\pi \\
                                  0 & 0 & \frac{2\sqrt{2}}{3}
               \end{pmatrix}
     

\end{align}

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N-dimensional Euclidean space If the Euclidean space is N-, and

$\backslash begin\{align\}$

 \mathbf{a} &= \sum_{i=1}^N a_i\mathbf{e}_i = a_1 \mathbf{e}_1 + a_2 \mathbf{e}_2 + {\ldots} + a_N \mathbf{e}_N \\
 \mathbf{b} &= \sum_{j=1}^N b_j\mathbf{e}_j = b_1 \mathbf{e}_1 + b_2 \mathbf{e}_2 + \ldots + b_N \mathbf{e}_N
     

\end{align} 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 a_i), then in algebraic form their dyadic product is:

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

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

$$

 \mathbf{ab} = \mathbf{ab}^\mathsf{T} =
 \begin{pmatrix}
      a_1 \\
      a_2 \\
   \vdots \\
      a_N
 \end{pmatrix}\begin{pmatrix}
   b_1 & b_2 & \cdots & b_N
 \end{pmatrix} =
 \begin{pmatrix}
   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{pmatrix}.
     

A dyadic polynomial A, otherwise known as a dyadic, is formed from multiple vectors a _i and b _j:

$$

 \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 + \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 (1983).

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Classification The following table classifies dyadics:

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Identities

The following identities are a direct consequence of the definition of the tensor product:Spencer (1992), page 19.

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Dyadic algebra

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Product of dyadic and vector

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

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Product of dyadic and dyadic

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

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Letting

$\backslash mathbf\{A\}\; =\; \backslash sum\_i\; \backslash mathbf\{a\}\_i\backslash mathbf\{b\}\_i,\backslash quad\; \backslash mathbf\{B\}\; =\; \backslash sum\_j\; \backslash mathbf\{c\}\_j\backslash mathbf\{d\}\_j$

be two general dyadics, we have:

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Double-dot product

The first definition of the double-dot product is the Frobenius inner product,

$$

\begin{align}

   \operatorname{tr}\left(\mathbf{A}\mathbf{B}^\mathsf{T}\right)
 &=\sum_{i,j} \operatorname{tr}\left(\mathbf{a}_i \mathbf{b}_i^\mathsf{T} \mathbf{d}_j \mathbf{c}_j^\mathsf{T}\right) \\
 &=\sum_{i,j} \operatorname{tr}\left(\mathbf{c}_j^\mathsf{T} \mathbf{a}_i \mathbf{b}_i^\mathsf{T} \mathbf{d}_j\right) \\
 &=\sum_{i,j} (\mathbf{a}_i\cdot\mathbf{c}_j)(\mathbf{b}_i\cdot\mathbf{d}_j) \\
 &=\mathbf{A} {}_\centerdot^\centerdot \mathbf{B}
     

\end{align}

Furthermore, since,

$$

\begin{align}

   \mathbf{A}^\mathsf{T}
 &=\sum_{i,j} \left(\mathbf{a}_i\mathbf{b}_j^\mathsf{T}\right)^\mathsf{T} \\
 &=\sum_{i,j} \mathbf{b}_i\mathbf{a}_j^\mathsf{T}
     

\end{align} we get that,

$$

\mathbf{A} {}_\centerdot^\centerdot \mathbf{B} = \mathbf{A} \underline, \hat{\mathbf{c}}, the unit dyadic is expressed by

$\backslash mathbf\{I\}\; =\; \backslash mathbf\{a\}\backslash hat\{\backslash mathbf\{a\}\}\; +\; \backslash mathbf\{b\}\backslash hat\{\backslash mathbf\{b\}\}\; +\; \backslash mathbf\{c\}\backslash hat\{\backslash mathbf\{c\}\}$

In the standard basis (for definitions of i, j, k see in the above section ),

$\backslash mathbf\{I\}\; =\; \backslash mathbf\{ii\}\; +\; \backslash mathbf\{jj\}\; +\; \backslash mathbf\{kk\}$

Explicitly, the dot product to the right of the unit dyadic is

$\backslash begin\{align\}$

\mathbf{I} \cdot \mathbf{a} & = (\mathbf{i}\mathbf{i} + \mathbf{j}\mathbf{j} + \mathbf{k}\mathbf{k})\cdot \mathbf{a} \\ & = \mathbf{i}(\mathbf{i} \cdot \mathbf{a}) + \mathbf{j}(\mathbf{j} \cdot \mathbf{a}) + \mathbf{k} (\mathbf{k} \cdot \mathbf{a}) \\ & = \mathbf{i} a_x + \mathbf{j} a_y + \mathbf{k} a_z \\ & = \mathbf{a} \end{align} and to the left

$\backslash begin\{align\}$

\mathbf{a} \cdot \mathbf{I} & = \mathbf{a} \cdot (\mathbf{i}\mathbf{i} + \mathbf{j}\mathbf{j} + \mathbf{k}\mathbf{k})\\ & = (\mathbf{a}\cdot \mathbf{i})\mathbf{i} + (\mathbf{a}\cdot \mathbf{j})\mathbf{j} + (\mathbf{a}\cdot \mathbf{k})\mathbf{k} \\ & = a_x \mathbf{i} + a_y \mathbf{j} + a_z \mathbf{k} \\ & = \mathbf{a} \end{align}

The corresponding matrix is

$\backslash mathbf\{I\}=\backslash begin\{pmatrix\}$

1 & 0 & 0\\
0 & 1 & 0\\
0 & 0 & 1\\
     

\end{pmatrix}

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 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 a₁ i + a₂ j + a₃ k to a₂ i, and jj sends this sum to a₂ j. Now it is revealed in what (precise) sense ii + jj + kk is the identity: it sends a₁ i + a₂ j + a₃ k 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.

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Properties of unit dyadics

$\backslash begin\{align\}$

 \left(\mathbf{a}\times\mathbf{I}\right)\cdot\left(\mathbf{b}\times\mathbf{I}\right)
   &= \mathbf{ba} - \left(\mathbf{a}\cdot\mathbf{b}\right)\mathbf{I} \\
 \mathbf{I} {}_\times^{\,\centerdot} \left(\mathbf{ab}\right)
   &= \mathbf{b}\times\mathbf{a} \\
 \mathbf{I} {}_\times^\times \mathbf{A}
   &= (\mathbf{A} {}_{\,\centerdot}^{\,\centerdot} \mathbf{I})\mathbf{I} - \mathbf{A}^\mathsf{T} \\
 \mathbf{I} {}_{\,\centerdot}^{\,\centerdot} \left(\mathbf{ab}\right)
   &= \left(\mathbf{I}\cdot\mathbf{a}\right)\cdot\mathbf{b}
    = \mathbf{a}\cdot\mathbf{b}
    = \mathrm{tr}\left(\mathbf{ab}\right)
     

\end{align} where "tr" denotes the trace.

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Examples

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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;

$\backslash mathbf\{a\}\; =\; \backslash mathbf\{a\}\_\backslash parallel\; +\; \backslash mathbf\{a\}\_\backslash perp$

The parallel component is found by vector projection, which is equivalent to the dot product of a with the dyadic nn,

$\backslash mathbf\{a\}\_\backslash parallel\; =\; \backslash mathbf\{n\}(\backslash mathbf\{n\}\backslash cdot\backslash mathbf\{a\})\; =\; (\backslash mathbf\{nn\})\backslash cdot\backslash mathbf\{a\}$

and the perpendicular component is found from vector rejection, which is equivalent to the dot product of a with the dyadic ,

$\backslash mathbf\{a\}\_\backslash perp\; =\; \backslash mathbf\{a\}\; -\; \backslash mathbf\{n\}(\backslash mathbf\{n\}\backslash cdot\backslash mathbf\{a\})\; =\; (\backslash mathbf\{I\}\; -\; \backslash mathbf\{nn\})\backslash cdot\backslash mathbf\{a\}$

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Rotation dyadic

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2d rotations

The dyadic

$\backslash mathbf\{J\}\; =\; \backslash mathbf\{ji\}\; -\; \backslash mathbf\{ij\}\; =\; \backslash begin\{pmatrix\}$

0 & -1 \\
1 & 0
     

\end{pmatrix} is a 90° anticlockwise rotation operator in 2d. It can be left-dotted with a vector r = x i + yj to produce the vector,

$(\backslash mathbf\{j\; i\}\; -\; \backslash mathbf\{i\; j\})\; \backslash cdot\; (x\; \backslash mathbf\{i\}\; +\; y\; \backslash mathbf\{j\})\; =$

x \mathbf{j i} \cdot \mathbf{i} - x \mathbf{i j} \cdot \mathbf{i} + y \mathbf{j i} \cdot \mathbf{j} - y \mathbf{i j} \cdot \mathbf{j} = -y \mathbf{i} + x \mathbf{j}, in summary

$\backslash mathbf\{J\}\backslash cdot\backslash mathbf\{r\}\; =\; \backslash mathbf\{r\}\_\backslash mathrm\{rot\}$

or in matrix notation

$$

\begin{pmatrix}

0 & -1 \\
1 & 0
     

\end{pmatrix} \begin{pmatrix}

x \\
y
     

\end{pmatrix}= \begin{pmatrix} -y \\

x
     

\end{pmatrix}.

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

$\backslash mathbf\{R\}\; =\; \backslash mathbf\{I\}\backslash cos\backslash theta\; +\; \backslash mathbf\{J\}\backslash sin\backslash theta\; =\; (\backslash mathbf\{ii\}+\backslash mathbf\{jj\})\backslash cos\backslash theta\; +\; (\backslash mathbf\{ji\}-\backslash mathbf\{ij\})\backslash sin\backslash theta\; =$

\begin{pmatrix} \cos\theta &-\sin\theta \\ \sin\theta &\;\cos\theta \end{pmatrix} where I and J are as above, and the rotation of any 2d vector a = a_x i + a_yj is

$\backslash mathbf\{a\}\_\backslash mathrm\{rot\}\; =\; \backslash mathbf\{R\}\backslash cdot\backslash mathbf\{a\}$

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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

$\backslash mathbf\{a\}\_\backslash mathrm\{rot\}\; =\; \backslash mathbf\{R\}\; \backslash cdot\; \backslash mathbf\{a\}\; \backslash ,,$

where the rotation dyadic is

$\backslash mathbf\{R\}\; =\; \backslash mathbf\{I\}\; \backslash cos\backslash theta\; +\; \backslash boldsymbol\{\backslash Omega\}\; \backslash sin\backslash theta\; +\; \backslash boldsymbol\{\backslash omega\backslash omega\}\; (1\; -\; \backslash cos\backslash theta)\; \backslash ,,$

and the Cartesian entries of ω also form those of the dyadic

$\backslash boldsymbol\{\backslash Omega\}\; =\; \backslash omega\_x(\; \backslash mathbf\{kj\}\; -\; \backslash mathbf\{jk\}\; )\; +\; \backslash omega\_y(\; \backslash mathbf\{ik\}\; -\; \backslash mathbf\{ki\}\; )\; +\; \backslash omega\_z(\; \backslash mathbf\{ji\}\; -\; \backslash mathbf\{ij\}\; )\; \backslash ,,$

The effect of Ω on a is the cross product

$\backslash boldsymbol\{\backslash Omega\}\; \backslash cdot\; \backslash mathbf\{a\}\; =\; \backslash boldsymbol\{\backslash omega\}\; \backslash times\; \backslash mathbf\{a\}$

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

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Lorentz transformation

In special relativity, the Lorentz boost with speed v in the direction of a unit vector n can be expressed as

$t\text{'}\; =\; \backslash gamma\backslash left(t\; -\; \backslash frac\{v\backslash mathbf\{n\}\backslash cdot\backslash mathbf\{r\}\}\{c^2\}\; \backslash right)$

$\backslash mathbf\{r\}\text{'}\; =\; \backslash mathbf\{I\}\backslash cdot\; \backslash mathbf\{r\}\; -\; \backslash gamma\; v\; \backslash mathbf\{n\}t$

where

$\backslash gamma=\backslash frac\{1\}\{\backslash sqrt\{1-\backslash dfrac\{v^2\}\{c^2\}\}\}$

is the Lorentz factor.

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Related terms

Some authors generalize from the term dyadic to related terms triadic, tetradic and polyadic.For example,

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See also

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

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Notes

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Explanatory notes

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Citations

• Chapter 2
• (2025). 9780071615457, McGraw Hill. ISBN 9780071615457
• A.J.M. Spencer (1992). 9780486435947, Dover Publications. ISBN 9780486435947
  .
• .
• Ismo V. Lindell (1996). 9780780360396, Wiley-Blackwell. ISBN 9780780360396
  .
• Hollis C. Chen (1983). 9780070106888, McGraw Hill. ISBN 9780070106888
  .
• K. Cahill (2025). 9781107005211, Cambridge University Press. ISBN 9781107005211

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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 PhD
• Advanced Field Theory, I.V.Lindel
• Vector and Dyadic Analysis
• Introductory Tensor Analysis
• Nasa.gov, Foundations of Tensor Analysis for students of Physics and Engineering with an Introduction to the Theory of Relativity, J.C. Kolecki
• Nasa.gov, An introduction to Tensors for students of Physics and Engineering, J.C. Kolecki

Categories: Tensors

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