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| In [[algebra]], the '''Binet–Cauchy identity''', named after [[Jacques Philippe Marie Binet]] and [[Augustin-Louis Cauchy]], states that <ref name=Weisstein>
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| {{cite book |title=CRC concise encyclopedia of mathematics |author=Eric W. Weisstein |page=228 |url=http://books.google.com/books?id=8LmCzWQYh_UC&pg=PA228 |chapter=Binet-Cauchy identity |isbn=1-58488-347-2 |year=2003 |edition=2nd |publisher=CRC Press}}
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| </ref>
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| : <math>
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| \biggl(\sum_{i=1}^n a_i c_i\biggr)
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| \biggl(\sum_{j=1}^n b_j d_j\biggr) =
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| \biggl(\sum_{i=1}^n a_i d_i\biggr)
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| \biggl(\sum_{j=1}^n b_j c_j\biggr)
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| + \sum_{1\le i < j \le n}
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| (a_i b_j - a_j b_i )
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| (c_i d_j - c_j d_i )
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| </math>
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| for every choice of [[real number|real]] or [[complex number]]s (or more generally, elements of a [[commutative ring]]).
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| Setting ''a<sub>i</sub>'' = ''c<sub>i</sub>'' and ''b<sub>j</sub>'' = ''d<sub>j</sub>'', it gives the [[Lagrange's identity]], which is a stronger version of the [[Cauchy–Schwarz inequality]] for the [[Euclidean space]] <math>\scriptstyle\mathbb{R}^n</math>.
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| ==The Binet–Cauchy identity and exterior algebra==
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| When ''n'' = 3 the first and second terms on the right hand side become the squared magnitudes of [[Dot product|dot]] and [[cross product]]s respectively; in ''n'' dimensions these become the magnitudes of the dot and [[wedge product]]s. We may write it
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| :<math>(a \cdot c)(b \cdot d) = (a \cdot d)(b \cdot c) + (a \wedge b) \cdot (c \wedge d)\,</math>
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| where '''a''', '''b''', '''c''', and '''d''' are vectors. It may also be written as a formula giving the dot product of two wedge products, as
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| :<math>(a \wedge b) \cdot (c \wedge d) = (a \cdot c)(b \cdot d) - (a \cdot d)(b \cdot c).\,</math>
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| In the special case of unit vectors ''a=c'' and ''b=d'', the formula yields
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| :<math>|a \wedge b|^2 = |a|^2|b|^2 - |a \cdot b|^2. \,</math>
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| When both vectors are unit vectors, we obtain the usual relation
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| :<math>1= \cos^2(\phi)+\sin^2(\phi)</math>
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| where φ is the angle between the vectors.
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| ==Proof==
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| Expanding the last term,
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| :<math>
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| \sum_{1\le i < j \le n}
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| (a_i b_j - a_j b_i )
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| (c_i d_j - c_j d_i )
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| </math>
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| :<math>
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| =
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| \sum_{1\le i < j \le n}
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| (a_i c_i b_j d_j + a_j c_j b_i d_i)
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| +\sum_{i=1}^n a_i c_i b_i d_i
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| -
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| \sum_{1\le i < j \le n}
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| (a_i d_i b_j c_j + a_j d_j b_i c_i)
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| -
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| \sum_{i=1}^n a_i d_i b_i c_i
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| </math>
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| where the second and fourth terms are the same and artificially added to complete the sums as follows:
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| :<math> | |
| =
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| \sum_{i=1}^n \sum_{j=1}^n
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| a_i c_i b_j d_j
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| -
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| \sum_{i=1}^n \sum_{j=1}^n
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| a_i d_i b_j c_j.
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| </math>
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| This completes the proof after factoring out the terms indexed by ''i''.
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| ==Generalization==
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| A general form, also known as the [[Cauchy–Binet formula]], states the following:
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| Suppose ''A'' is an ''m''×''n'' [[matrix (mathematics)|matrix]] and ''B'' is an ''n''×''m'' matrix. If ''S'' is a [[subset]] of {1, ..., ''n''} with ''m'' elements, we write ''A<sub>S</sub>'' for the ''m''×''m'' matrix whose columns are those columns of ''A'' that have indices from ''S''. Similarly, we write ''B<sub>S</sub>'' for the ''m''×''m'' matrix whose ''rows'' are those rows of ''B'' that have indices from ''S''.
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| Then the [[determinant]] of the [[matrix product]] of ''A'' and ''B'' satisfies the identity
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| :<math>\det(AB) = \sum_{\scriptstyle S\subset\{1,\ldots,n\}\atop\scriptstyle|S|=m} \det(A_S)\det(B_S),</math>
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| where the sum extends over all possible subsets ''S'' of {1, ..., ''n''} with ''m'' elements.
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| We get the original identity as special case by setting
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| :<math>
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| A=\begin{pmatrix}a_1&\dots&a_n\\b_1&\dots& b_n\end{pmatrix},\quad
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| B=\begin{pmatrix}c_1&d_1\\\vdots&\vdots\\c_n&d_n\end{pmatrix}.
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| </math>
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| ==In-line notes and references==
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| <references/>
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| {{DEFAULTSORT:Binet-Cauchy Identity}}
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| [[Category:Mathematical identities]]
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| [[Category:Multilinear algebra]]
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| [[Category:Articles containing proofs]]
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