dyadic tensor

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}

The identity dyadic tensor in three dimensions is

i i + j j + k k.

The dyadic tensor

j i − i j

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}.

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