Dyadic tensor

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A dyadic tensor in multilinear algebra is a second rank tensor written in a special notation, formed by juxtaposing pairs of vectors, i.e. placing pairs of vectors side by side.

Each component of a dyadic tensor is a dyad. A dyad is the juxtaposition of a pair of basis vectors and a scalar coefficient.

As an example, let

\mathbf{A} = a \mathbf{i} + b \mathbf{j}

and

\mathbf{X} = x \mathbf{i} + y \mathbf{j}

be a pair of two-dimensional vectors. Then the juxtaposition of A and X is

\mathbf{A X} = a x \mathbf{i i} + a y \mathbf{i j} + b x \mathbf{j i} + b y \mathbf{j j} = \left( \begin{array}{cc} \text{ax} & \text{ay} \\ \text{bx} & \text{by} \end{array} \right).

The identity dyadic tensor in three dimensions is

I=i i + j j + k k

The dyadic tensor

J=j i − i j = \left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)

is a 90° rotation operator in two dimensions. It can be dotted (from the left) with a vector to produce the rotation:

(\mathbf{j i} - \mathbf{i j}) \cdot (x \mathbf{i} + y \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}.
\left( \begin{array}{cc} 0 & -1 \\ 1 & 0 \end{array} \right)\left( \begin{array}{c} x \\ y \end{array} \right)=\left( \begin{array}{c} -y \\ x \end{array} \right)

a General 2-D Rotation Dyadic for θ angle, anti-clockwise

I \text{cos}[\theta ] + J \text{sin}[\theta ]=\left( \begin{array}{cc} \text{cos}[\theta ] & -\text{sin}[\theta ] \\ \text{sin}[\theta ] & \text{cos}[\theta ] \end{array} \right)

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.

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