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Dyadic product
From Citizendium, the Citizens' Compendium
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 U ⊗ V be their tensor product space
If {a_{i}} and {b_{j}} are bases of U and V, respectively, then
and
The dyadic product u ⊗ v is an m × n matrix that represents the simple tensor u ⊗ v in U ⊗ V.