Dyadic product

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In mathematics, a dyadic product of two vectors is a third vector product next to dot product and cross product. The dyadic product is a square matrix that represents a tensor with respect to the same system of axes as to which the components of the vectors are defined that constitute the dyadic product. Thus, if

then the dyadic product is

Sometimes it is useful to write a dyadic product as matrix product of two matrices, the first being a column matrix and the second a row matrix,

Example

Use

An important use of a dyadic product is the reformulation of a vector expression as a matrix-vector expression, for instance,

Indeed, take the ith component,

Or, equivalently, by use of the associative law valid for matrix multiplication,

Multiplication

The matrix multiplication of two dyadic products is given by,

Generalization

In more general terms, a dyadic product is the representation of a simple element in (binary) tensor product space with respect to bases carrying the constituting spaces. Let U and V be linear spaces and UV be their tensor product space

If {ai} and {bj} are bases of U and V, respectively, then

and

The dyadic product uv is an m × n matrix that represents the simple tensor uv in UV.