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| :''For "outer product" in [[geometric algebra]], see [[Exterior algebra|exterior product]].''
| | The use of ecigs can be a exercise that's becoming increasingly popular, and many people have taken up the utilization of e-cigs as a solution option to smoking. It is because e-cigs do not bring about ash, plus they do not cause a person’s property, garments, and auto stinking poor. One of the biggest parts of needs to employ e-cigs involves purchasing a starter kit, and there are a quantity of factors an individual should be aware of about purchasing a beginner kit. Including purchasing a marginally greater beginner kit, and there are always a handful of things someone needs to have a part of their starter kit.<br><br>Investing in a somewhat greater kit<br><br>One of the most typical blunders that persons create when first starting to smoke e-cigs is the fact that they tend to purchase the cheapest ecig that is available to them. This could cause a individual certainly not trying out the practice, and it also implies that an individual will never purchase a wonderful e cig when they do use up the practice. This is the reason it's recommended to purchase a better one, that looks just how a person want to buy to, and it's also a good good thought to purchase one which can be gradually improved with time.<br><br>What ought to be contained in the kit<br><br>A good starter-kit may have such things as a charger that is user friendly. The kit should also contain guidelines on how best to populate the e-cigarette with liquid, a rag to scrub it with, also it also needs to possess a some additional e-juice or capsules. Visit [http://ciglites.metroblog.com/ [http://ciglites.metroblog.com/ Read More On this page]]. |
| In [[linear algebra]], the '''outer product''' typically refers to the [[tensor product]] of two [[vector (mathematics)|vectors]]. The result of applying the outer product to a pair of [[coordinate vector]]s is a [[matrix (mathematics)|matrix]]. The name contrasts with the [[inner product]], which takes as input a pair of vectors and produces a [[scalar (mathematics)|scalar]].
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| The outer product of vectors can be also regarded as a special case of the [[Kronecker product]] of matrices.
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| Some authors use the expression "outer product of tensors" as a synonym of "tensor product". The outer product is also a [[higher-order function]] in some computer programming languages such as [[APL programming language|APL]] and [[Mathematica]].
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| ==Definition (matrix multiplication)==
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| {{main|matrix multiplication}}
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| The outer product {{nowrap|'''u''' ⊗ '''v'''}} is equivalent to a matrix multiplication '''uv'''<sup>T</sup>, provided that '''u''' is represented as a {{nowrap|''m'' × 1}} [[column vector]] and '''v''' as a {{nowrap|''n'' × 1}} column vector (which makes '''v'''<sup>T</sup> a row vector).<ref>Linear Algebra (4th Edition), S. Lipcshutz, M. Lipson, Schaum’s Outlines, McGraw Hill (USA), 2009, ISBN 978-0-07-154352-1</ref> For instance, if {{nowrap|1=''m'' = 4}} and {{nowrap|1=''n'' = 3}}, then
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| :<math>\mathbf{u} \otimes \mathbf{v} = \mathbf{u} \mathbf{v}^\mathrm{T} =
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| \begin{bmatrix}u_1 \\ u_2 \\ u_3 \\ u_4\end{bmatrix}
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| \begin{bmatrix}v_1 & v_2 & v_3\end{bmatrix} =
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| \begin{bmatrix}u_1v_1 & u_1v_2 & u_1v_3 \\ u_2v_1 & u_2v_2 & u_2v_3 \\ u_3v_1 & u_3v_2 & u_3v_3 \\ u_4v_1 & u_4v_2 & u_4v_3\end{bmatrix}.</math>
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| For [[complex numbers|complex]] vectors, it is customary to use the [[conjugate transpose]] of '''v''' (denoted '''v'''<sup>H</sup>):
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| :<math>\mathbf{u} \otimes \mathbf{v} = \mathbf{u} \mathbf{v}^\mathrm{H}.</math>
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| ===Contrast with inner product===
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| If {{nowrap|1=''m'' = ''n''}}, then one can take the matrix product the other way, yielding a scalar (or {{nowrap|1 × 1}} matrix):
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| :<math>\left\langle \mathbf{u}, \mathbf{v}\right\rangle = \mathbf{v}^\mathrm{H} \mathbf{u}</math>
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| which is the standard [[inner product]] for [[Euclidean vector space]]s, better known as the [[dot product]]. The inner product is the [[trace (linear algebra)|trace]] of the outer product.
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| ==Definition (vectors and tensors)==
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| ===Vector multiplication===
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| Given the vectors
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| :<math>\begin{align}
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| \mathbf{u} & =(u_1, u_2, \dots, u_m) \\
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| \mathbf{v} & = (v_1, v_2, \dots, v_n)
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| \end{align}</math>
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| their outer product {{nowrap|'''u''' ⊗ '''v'''}} is defined as the {{nowrap|''m'' × ''n''}} matrix '''A''' obtained by multiplying each element of '''u''' by each element of '''v''':<ref>http://mathworld.wolfram.com/KroneckerProduct.html</ref><ref>Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.) 0-89573-752-3
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| </ref>
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| :<math>\mathbf{u} \otimes \mathbf{v} = \mathbf{A} =
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| \begin{bmatrix}u_1v_1 & u_1v_2 & \dots & u_1v_n \\ u_2v_1 & u_2v_2 & \dots & u_2v_n \\ \vdots & \vdots & \ddots & \vdots\\ u_mv_1 & u_mv_2 & \dots & u_mv_n \end{bmatrix}.</math>
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| For complex vectors, the [[complex conjugate]] of '''v''' (denoted '''v'''<sup>∗</sup> or '''v̅'''). Namely, matrix '''A''' is obtained by multiplying each element of '''u''' by the complex conjugate of each element of '''v'''.
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| ===Tensor multiplication===
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| The outer product on tensors is typically referred to as the [[tensor product]]. Given a [[tensor]] '''a''' with [[Tensor rank|rank]] ''q'' and [[Dimension (vector space)|dimension]]s {{nowrap|(''i''<sub>1</sub>, ..., ''i''<sub>''q''</sub>)}}, and a tensor '''b''' with rank ''r'' and dimensions {{nowrap|(''j''<sub>1</sub>, ..., ''j''<sub>''r''</sub>)}}, their outer product '''c''' has rank {{nowrap|''q'' + ''r''}} and dimensions {{nowrap|(''k''<sub>'''1'''</sub>, ..., ''k''<sub>''q''+''r''</sub>)}} which are the ''i'' dimensions followed by the ''j'' dimensions. It is denoted in coordinate-free notation using ⊗ and components are defined [[index notation]] by:<ref>Mathematical methods for physics and engineering, K.F. Riley, M.P. Hobson, S.J. Bence, Cambridge University Press, 2010, ISBN 978-0-521-86153-3</ref>
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| :<math>\mathbf{c}=\mathbf{a}\otimes\mathbf{b}, \quad c_{ij}=a_ib_j </math>
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| similarly for higher order tensors:
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| :<math>\mathbf{T}=\mathbf{a}\otimes\mathbf{b}\otimes\mathbf{c}, \quad T_{ijk}=a_ib_jc_k </math>
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| For example, if '''A''' has rank 3 and dimensions {{nowrap|(3, 5, 7)}} and '''B''' has rank 2 and dimensions {{nowrap|(10, 100)}}, their outer product '''c''' has rank 5 and dimensions {{nowrap|(3, 5, 7, 10, 100)}}. If '''A''' has a component {{nowrap|1=''A''<sub>[2, 2, 4]</sub> = 11}} and '''B''' has a component {{nowrap|1=''B''<sub>[8, 88]</sub> = 13}}, then the component of '''C''' formed by the outer product is {{nowrap|1=''C''<sub>[2, 2, 4, 8, 88]</sub> = 143}}.
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| To understand the matrix definition of outer product in terms of the definition of tensor product:
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| # The vector '''v''' can be interpreted as a rank 1 tensor with dimension ''M'', and the vector '''u''' as a rank 1 tensor with dimension ''N''. The result is a rank 2 tensor with dimension {{nowrap|(''M'', ''N'')}}.
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| # The rank of the result of an [[inner product]] between two tensors of rank ''q'' and ''r'' is the greater of {{nowrap|''q'' + ''r'' − 2}} and 0. Thus, the inner product of two matrices has the same rank as the outer product (or tensor product) of two vectors.
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| # It is possible to add arbitrarily many leading or trailing ''1'' dimensions to a tensor without fundamentally altering its structure. These ''1'' dimensions would alter the character of operations on these tensors, so any resulting equivalences should be expressed explicitly.
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| # The inner product of two matrices '''V''' with dimensions {{nowrap|(''d'', ''e'')}} and '''U''' with dimensions {{nowrap|(''e'', ''f'')}} is <math>\sum_{j = 1}^e V_{ij} U_{jk}</math>, where {{nowrap|1=''i'' = 1, 2, ..., ''d''}} and {{nowrap|1=''k'' = 1, 2, ..., ''f''}}. For the case where {{nowrap|1=''e'' = 1}}, the summation is trivial (involving only a single term).
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| # The outer product of two matrices '''V''' with dimensions {{nowrap|(''m'', ''n'')}} and '''U''' with dimensions {{nowrap|(''p'', ''q'')}} is <math> C_{st} = V_{ij} U_{hk}</math>, where {{nowrap|1=''s'' = 1, 2, ..., ''mp'' − 1, ''mp''}} and {{nowrap|1=''t'' = 1, 2, ..., ''nq'' − ''1'', ''nq''}}.
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| The term "rank" is used here in its [[tensor]] sense, and should not be interpreted as [[Rank (linear algebra)|matrix rank]].
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| ==Definition (abstract)==
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| Let ''V'' and ''W'' be two [[vector space]]s, and let ''W''<sup>∗</sup> be the [[dual space]] of ''W''.
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| Given a vector {{nowrap|''x'' ∈ ''V''}} and {{nowrap|''y''<sup>∗</sup> ∈ ''W''<sup>∗</sup>}}, then the tensor product {{nowrap|''y''<sup>∗</sup> ⊗ ''x''}} corresponds to the map {{nowrap|''A'' : W → ''V''}} given by
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| :<math>w \mapsto y^*(w)x.</math>
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| Here ''y''<sup>∗</sup>(''w'') denotes the value of the [[linear functional]] ''y''<sup>∗</sup> (which is an element of the dual space of ''W'') when evaluated at the element {{nowrap|''w'' ∈ ''W''}}. This scalar in turn is multiplied by ''x'' to give as the final result an element of the space ''V''.
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| If ''V'' and ''W'' are finite-dimensional, then the space of all linear transformations from ''W'' to ''V'', denoted {{nowrap|Hom(''W'', ''V'')}}, is generated by such outer products; in fact, the rank of a matrix is the minimal number of such outer products needed to express it as a sum (this is the '''tensor rank''' of a matrix). In this case {{nowrap|Hom(''W'', ''V'')}} is [[isomorphic]] to {{nowrap|''W''<sup>∗</sup> ⊗ ''V''}}.
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| ===Contrast with inner product===
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| {{See also|Inner product space}}
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| If {{nowrap|1=''W'' = ''V''}}, then one can also pair the covector {{nowrap|''w''<sup>∗</sup> ∈ ''V''<sup>∗</sup>}} with the vector {{nowrap|''v'' ∈ ''V''}} via {{nowrap|(''w''<sup>∗</sup>, ''v'') → ''w''<sup>∗</sup>(''v'')}}, which is the duality pairing between ''V'' and its dual, sometimes called the [[inner product]].
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| ==Applications==
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| The outer product is useful in computing physical quantities (e.g., the [[Moment of inertia|tensor of inertia]]), and performing transform operations in [[digital signal processing]] and [[digital image processing]]. It is also useful in [[statistical analysis]] for computing the [[covariance]] and auto-covariance matrices for two [[random variables]].
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| ==See also==
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| * [[Linear algebra]]
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| * [[Norm (mathematics)]]
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| * [[Scatter matrix]]
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| * [[Ricci calculus]]
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| ===Products===
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| * [[Cross product]]
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| * [[Exterior product]]
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| ===Duality===
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| * [[Complex conjugate]]
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| * [[Conjugate transpose]]
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| * [[Transpose]]
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| * [[Bra–ket notation#Outer products|Bra–ket notation for outer product]]
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| ==References==
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| {{reflist}}
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| {{Linear algebra}}
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| {{DEFAULTSORT:Outer Product}}
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| [[Category:Bilinear operators]]
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| [[Category:Binary operations]]
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| [[Category:Higher-order functions]]
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The use of ecigs can be a exercise that's becoming increasingly popular, and many people have taken up the utilization of e-cigs as a solution option to smoking. It is because e-cigs do not bring about ash, plus they do not cause a person’s property, garments, and auto stinking poor. One of the biggest parts of needs to employ e-cigs involves purchasing a starter kit, and there are a quantity of factors an individual should be aware of about purchasing a beginner kit. Including purchasing a marginally greater beginner kit, and there are always a handful of things someone needs to have a part of their starter kit.
Investing in a somewhat greater kit
One of the most typical blunders that persons create when first starting to smoke e-cigs is the fact that they tend to purchase the cheapest ecig that is available to them. This could cause a individual certainly not trying out the practice, and it also implies that an individual will never purchase a wonderful e cig when they do use up the practice. This is the reason it's recommended to purchase a better one, that looks just how a person want to buy to, and it's also a good good thought to purchase one which can be gradually improved with time.
What ought to be contained in the kit
A good starter-kit may have such things as a charger that is user friendly. The kit should also contain guidelines on how best to populate the e-cigarette with liquid, a rag to scrub it with, also it also needs to possess a some additional e-juice or capsules. Visit [http://ciglites.metroblog.com/ Read More On this page].