Dyadic tensor
From Free net encyclopedia
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
- <math> \mathbf{A} = a \mathbf{i} + b \mathbf{j} </math>
and
- <math> \mathbf{X} = x \mathbf{i} + y \mathbf{j} </math>
be a pair of two-dimensional vectors. Then the juxtaposition of A and X is
- <math> \mathbf{A X} = a x \mathbf{i i} + a y \mathbf{i j} + b x \mathbf{j i} + b y \mathbf{j j} </math>.
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:
- <math> (\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}.</math>