Dyadic product: Difference between revisions
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{{mergeto|tensor product|date=August 2012|discuss=Wikipedia talk:WikiProject Mathematics#Suggested merges with dyadic product and outer product, into tensor product...}} |
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In [[mathematics]], in particular [[multilinear algebra]], the '''dyadic product''' |
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:<math>\mathbb{P} = \mathbf{u}\otimes\mathbf{v}</math> |
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of two [[Vector (geometric)|vector]]s, <math>\mathbf{u}</math> and <math>\mathbf{v}</math>, each having the same dimension, is the [[tensor product]] of the vectors and results in a [[tensor]] of [[Tensor order#Tensor rank|order]] two and [[Tensor#Tensor rank|rank]] one. It is also called [[outer product]]. |
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== Components == |
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With respect to a chosen [[Basis (linear algebra)|basis]] <math>\{\mathbf{e}_i\}</math>, the components <math>P_{ij}</math> of the dyadic product <math>\mathbb{P} = \mathbf{u} \otimes \mathbf{v}</math> may be defined by |
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:<math>\displaystyle P_{ij} = u_i v_j </math> , |
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where |
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:<math>\mathbf{u} = \sum_i u_i \mathbf{e}_i</math> , |
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:<math>\mathbf{v} = \sum_j v_j \mathbf{e}_j</math> , |
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and |
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:<math>\mathbb{P} = \sum_{i,j} P_{ij} \mathbf{e}_i \otimes \mathbf{e}_j</math> . |
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==Matrix representation== |
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The dyadic product can be simply represented as the square [[Matrix (mathematics)|matrix]] obtained by [[matrix multiplication|multiplying]] <math>\mathbf{u}</math> as a [[column vector]] by <math>\mathbf{v}</math> as a [[row vector]]. For example, |
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:<math> |
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\mathbf{u} \otimes \mathbf{v} |
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\rightarrow |
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\begin{bmatrix} |
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u_1 \\ |
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u_2 \\ |
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u_3 \end{bmatrix} |
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\begin{bmatrix} v_1 & v_2 & v_3 \end{bmatrix} |
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= |
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\begin{bmatrix} |
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u_1v_1 & u_1v_2 & u_1v_3 \\ |
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u_2v_1 & u_2v_2 & u_2v_3 \\ |
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u_3v_1 & u_3v_2 & u_3v_3 |
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\end{bmatrix} , |
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</math> |
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where the arrow indicates that this is only one particular representation of the dyadic product, referring to a particular [[basis (linear algebra)|basis]]. In this representation, the dyadic product is a special case of the [[Kronecker product]]. |
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==Identities== |
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The following identities are a direct consequence of the definition of the dyadic product<ref>See Spencer (1992), page 19.</ref>: |
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:<math> |
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\begin{align} |
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(\alpha \mathbf{u}) \otimes \mathbf{v} &= \mathbf{u} \otimes (\alpha \mathbf{v}) = \alpha (\mathbf{u} \otimes \mathbf{v}), \\ |
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\mathbf{u} \otimes (\mathbf{v} + \mathbf{w}) &= \mathbf{u} \otimes \mathbf{v} + \mathbf{u} \otimes \mathbf{w}, \\ |
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(\mathbf{u} + \mathbf{v}) \otimes \mathbf{w} &= \mathbf{u} \otimes \mathbf{w} + \mathbf{v} \otimes \mathbf{w}, \\ |
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(\mathbf{u} \otimes \mathbf{v}) \cdot \mathbf{w} &= \mathbf{u}\; (\mathbf{v} \cdot \mathbf{w}), \\ |
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\mathbf{u} \cdot (\mathbf{v} \otimes \mathbf{w}) &= (\mathbf{u} \cdot \mathbf{v})\; \mathbf{w}. |
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\end{align} |
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</math> |
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==See also== |
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* [[Dyadic tensor]] |
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* [[Tensor product]] |
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* [[Kronecker product]] |
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* [[Outer product]] |
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==Notes== |
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{{reflist}} |
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==References== |
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*{{cite book | title=Continuum Mechanics | author=A.J.M. Spencer | year=1992 | publisher=Dover Publications | isbn=0-486-43594-6 }}. |
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[[Category:Tensors]] |
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[[de:Dyadisches Produkt]] |
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[[pl:Iloczyn diadyczny]] |
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[[ru:Умножение двухэлементного тензора]] |
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[[sq:Produkti diadik]] |
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[[uk:Множення двохелементного тензора]] |
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[[zh:并矢积]] |
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Latest revision as of 23:20, 23 August 2012
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