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
and
be a pair of two-dimensional vectors. Then the juxtaposition of A and X is
.
The identity dyadic tensor in three dimensions is
- I=i i + j j + k k
The dyadic tensor
- J=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:
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)](../../../../math/1/6/8/168847aa911a708d8e5732f043ec07b9.png)
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.
