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| :''For "outer product" in [[geometric algebra]], see [[Exterior algebra|exterior product]].''
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| 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. | | The Wildcats' rebuilding program has hit a snag due the lack of offensive firepower. Now they draw just one of the West's toughest out-of-conference mid-major-type programs on the inside Cougars and desperately need a win or Arizona fans will be made even more restless before Pac-10 run.<br><br>In June 2011, I offered up this rose as a 'current favourite' and a single I was trialling my garden. I originally chose it because the plan was reported to be'.extremely healthy, repeat flowering all summer [if you dead head], almost thornless with clusters of pretty, single flowers starting as soft apricot buds opening to white with a hint of sentimental lemon and ideal for low hedges or within a mixed border'. A year later I will wholeheartedly recommend this pretty little rose, it's began on my list of 'good doers'.<br><br>Before traversing to a lawn and garden center to pick up plants and landscaping materials, take the time to get exact measurements of process area. It is then much in order to determine how much of each item you have the need for. This helps you not to ever waste money gas and time by ordering too much and needing to return the program.<br><br>You'd be blown away at how service station . get formerly their old, musty shower curtains, but this may be the first thing a buyer will notice upon entering the bath. Shower curtains are relatively cheap and they can enhance the risk for room appear brighter.<br><br>Your lawn can be studied care of once full week with a landscaper. You hire the do a number of different designs in your front design. Cross mowing, checkering, lines or maybe just plain buttoning a shirt are various different designs so that you can pick creating your grass look in top shape. A landscaper will rake all of the chopped grass for you so your roots don't die.<br><br>We brought all explanation accoutrements with us to the beach: umbrella, towels, magazines and a cooler along with drinks. One difference was that the cooler was filled with ice-cold home brew. Haulover Beach is not only lax around the usual clothing laws, it's also lax on the laws prohibiting alcoholic beverages on public beaches. For me, consider the 63 as good a reason as any to uncover what nudism centered on.<br><br>Those blades spin around very fast and will throw up any debris that is caught in them. For this reason just a few ingredients to specific they are evident of sticks and stones before you begin the serp.<br><br>In the event you adored this article as well as you want to acquire more information regarding [http://www.hopesgrovenurseries.co.uk/ hopesgrovenurseries.co.uk] generously go to our own web site. |
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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 Wildcats' rebuilding program has hit a snag due the lack of offensive firepower. Now they draw just one of the West's toughest out-of-conference mid-major-type programs on the inside Cougars and desperately need a win or Arizona fans will be made even more restless before Pac-10 run.
In June 2011, I offered up this rose as a 'current favourite' and a single I was trialling my garden. I originally chose it because the plan was reported to be'.extremely healthy, repeat flowering all summer [if you dead head], almost thornless with clusters of pretty, single flowers starting as soft apricot buds opening to white with a hint of sentimental lemon and ideal for low hedges or within a mixed border'. A year later I will wholeheartedly recommend this pretty little rose, it's began on my list of 'good doers'.
Before traversing to a lawn and garden center to pick up plants and landscaping materials, take the time to get exact measurements of process area. It is then much in order to determine how much of each item you have the need for. This helps you not to ever waste money gas and time by ordering too much and needing to return the program.
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We brought all explanation accoutrements with us to the beach: umbrella, towels, magazines and a cooler along with drinks. One difference was that the cooler was filled with ice-cold home brew. Haulover Beach is not only lax around the usual clothing laws, it's also lax on the laws prohibiting alcoholic beverages on public beaches. For me, consider the 63 as good a reason as any to uncover what nudism centered on.
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