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In mathematics , in particular multilinear algebra , the dyadic product
P
=
u
⊗
v
{\displaystyle \mathbb {P} =\mathbf {u} \otimes \mathbf {v} }
of two vectors ,
u
{\displaystyle \mathbf {u} }
v
{\displaystyle \mathbf {v} }
tensor product of the vectors and results in a tensor of order two and rank one. It is also called outer product .
Components With respect to a chosen basis
{
e
i
}
{\displaystyle \{\mathbf {e} _{i}\}}
P
i
j
{\displaystyle P_{ij}}
P
=
u
⊗
v
{\displaystyle \mathbb {P} =\mathbf {u} \otimes \mathbf {v} }
P
i
j
=
u
i
v
j
{\displaystyle \displaystyle P_{ij}=u_{i}v_{j}}
where
u
=
∑
i
u
i
e
i
{\displaystyle \mathbf {u} =\sum _{i}u_{i}\mathbf {e} _{i}}
v
=
∑
j
v
j
e
j
{\displaystyle \mathbf {v} =\sum _{j}v_{j}\mathbf {e} _{j}}
and
P
=
∑
i
,
j
P
i
j
e
i
⊗
e
j
{\displaystyle \mathbb {P} =\sum _{i,j}P_{ij}\mathbf {e} _{i}\otimes \mathbf {e} _{j}}
Matrix representation The dyadic product can be simply represented as the square matrix obtained by multiplying
u
{\displaystyle \mathbf {u} }
column vector by
v
{\displaystyle \mathbf {v} }
row vector . For example,
u
⊗
v
→
[
u
1
u
2
u
3
]
[
v
1
v
2
v
3
]
=
[
u
1
v
1
u
1
v
2
u
1
v
3
u
2
v
1
u
2
v
2
u
2
v
3
u
3
v
1
u
3
v
2
u
3
v
3
]
,
{\displaystyle \mathbf {u} \otimes \mathbf {v} \rightarrow {\begin{bmatrix}u_{1}\\u_{2}\\u_{3}\end{bmatrix}}{\begin{bmatrix}v_{1}&v_{2}&v_{3}\end{bmatrix}}={\begin{bmatrix}u_{1}v_{1}&u_{1}v_{2}&u_{1}v_{3}\\u_{2}v_{1}&u_{2}v_{2}&u_{2}v_{3}\\u_{3}v_{1}&u_{3}v_{2}&u_{3}v_{3}\end{bmatrix}},}
where the arrow indicates that this is only one particular representation of the dyadic product, referring to a particular basis . In this representation, the dyadic product is a special case of the Kronecker product .
Identities The following identities are a direct consequence of the definition of the dyadic product[1]
(
α
u
)
⊗
v
=
u
⊗
(
α
v
)
=
α
(
u
⊗
v
)
,
u
⊗
(
v
+
w
)
=
u
⊗
v
+
u
⊗
w
,
(
u
+
v
)
⊗
w
=
u
⊗
w
+
v
⊗
w
,
(
u
⊗
v
)
⋅
w
=
u
(
v
⋅
w
)
,
u
⋅
(
v
⊗
w
)
=
(
u
⋅
v
)
w
.
{\displaystyle {\begin{aligned}(\alpha \mathbf {u} )\otimes \mathbf {v} &=\mathbf {u} \otimes (\alpha \mathbf {v} )=\alpha (\mathbf {u} \otimes \mathbf {v} ),\\\mathbf {u} \otimes (\mathbf {v} +\mathbf {w} )&=\mathbf {u} \otimes \mathbf {v} +\mathbf {u} \otimes \mathbf {w} ,\\(\mathbf {u} +\mathbf {v} )\otimes \mathbf {w} &=\mathbf {u} \otimes \mathbf {w} +\mathbf {v} \otimes \mathbf {w} ,\\(\mathbf {u} \otimes \mathbf {v} )\cdot \mathbf {w} &=\mathbf {u} \;(\mathbf {v} \cdot \mathbf {w} ),\\\mathbf {u} \cdot (\mathbf {v} \otimes \mathbf {w} )&=(\mathbf {u} \cdot \mathbf {v} )\;\mathbf {w} .\end{aligned}}}
See also Notes
^ See Spencer (1992), page 19.
References A.J.M. Spencer (1992). Continuum Mechanics . Dover Publications. ISBN 0486435946 
