Integrated Encryption Scheme - Revision history

en>Forderud: /* References */ add reference to survey article

2014-09-24T08:26:30Z

References: add reference to survey article

← Older revision		Revision as of 10:26, 24 September 2014
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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>~~		Hello, my name is Andrew and my wife doesn't like it at all. She functions as a travel agent but soon she'll be on her personal. To climb is some thing she would never give up. I've always cherished residing in Alaska.<br><br>Feel free to visit my website :: phone psychic ([http://fashionlinked.com/index.php?do=/profile-13453/info/ http://fashionlinked.com/])

	{{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}}

	~~</ref>~~

	~~: <math>~~
	~~\biggl(\sum_{i=1}^n a_i c_i\biggr)~~
	~~\biggl(\sum_{j=1}^n b_j d_j\biggr) =~~
	~~\biggl(\sum_{i=1}^n a_i d_i\biggr)~~
	~~\biggl(\sum_{j=1}^n b_j c_j\biggr)~~
	~~+ \sum_{1\le i < j \le n}~~
	~~(a_i b_j - a_j b_i )~~
	~~(c_i d_j - c_j d_i )~~
	~~</math>~~

	~~for every choice of [[real number\|real]] or [[complex number]]s (or more generally, elements of a [[commutative ring]]).~~
	~~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>.~~

	~~==The Binet–Cauchy identity and exterior algebra==~~
	~~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

	~~:<math>(a \cdot c)(b \cdot d) = (a \cdot d)(b \cdot c) + (a \wedge b) \cdot (c \wedge d)\,</math>~~

	~~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

	~~:<math>(a \wedge b) \cdot (c \wedge d) = (a \cdot c)(b \cdot d) - (~~a ~~\cdot d)(b \cdot c).\,</math>~~
	~~In the special case of unit vectors~~ '~~'a=c'' and ''b=d'', the formula yields~~
	~~:<math>\|a \wedge b\|^2 = \|a\|^2\|b\|^2 - \|a \cdot b\|^2~~. ~~\,</math>~~

	~~When both vectors are unit vectors, we obtain the usual relation~~
	~~:<math>1= \cos^2(\phi)+\sin^2(\phi)</math>~~
	~~where φ~~ is ~~the angle between the vectors~~.

	~~==Proof==~~
	~~Expanding the last term,~~

	:<~~math~~>
	~~\sum_{1\le i~~ < ~~j \le n}~~
	~~(a_i b_j - a_j b_i )~~
	~~(c_i d_j - c_j d_i )~~
	~~</math~~>
	~~:<math>~~
	=
	~~\sum_{1\le i < j \le n}~~
	~~(a_i c_i b_j d_j + a_j c_j b_i d_i)~~
	~~+\sum_{i=1}^n a_i c_i b_i d_i~~
	-
	~~\sum_{1\le i < j \le n}~~
	~~(a_i d_i b_j c_j + a_j d_j b_i c_i)~~
	-
	~~\sum_{i=1}^n a_i d_i b_i c_i~~
	~~</math>~~

	~~where the second and fourth terms are the same and artificially added~~ to ~~complete the sums as follows~~:

	:~~<math>~~
	=
	~~\sum_{i=1}^n \sum_{j=1}^n~~
	~~a_i c_i b_j d_j~~
	-
	~~\sum_{i=1}^n \sum_{j=1}^n~~
	~~a_i d_i b_j c_j.~~
	~~</math>~~

	~~This completes the proof after factoring out the terms indexed by ''i''.~~

	~~==Generalization==~~
	~~A general form, also known as the~~ [~~[Cauchy–Binet formula]], states the following~~:
	~~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''~~.
	~~Then the [[determinant]] of the [[matrix product]] of ''A'' and ''B'' satisfies the identity~~
	~~:<math>\det(AB) = \sum_{\scriptstyle S\subset\{1,\ldots,n\}\atop\scriptstyle\|S\|=m} \det(A_S)\det(B_S),<~~/~~math>~~
	~~where the sum extends over all possible subsets ''S'' of {1,~~ .~~.., ''n''} with ''m'' elements.~~

	~~We get the original identity as special case by setting~~
	~~:<math>~~
	A=~~\begin{pmatrix}a_1&\dots&a_n\\b_1&\dots& b_n\end{pmatrix},\quad~~
	~~B=\begin{pmatrix}c_1&d_1\\\vdots&\vdots\\c_n&d_n\end{pmatrix}.~~
	</~~math>~~

	~~==In~~-~~line notes and references==~~
	~~<references~~/>

	~~{{DEFAULTSORT~~:~~Binet-Cauchy Identity}}~~
	~~[[Category:Mathematical identities]]~~
	~~[[Category:Multilinear algebra]]~~
	~~[[Category:Articles containing proofs]~~]

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q1332372

2013-03-15T01:07:46Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q1332372

← Older revision		Revision as of 03:07, 15 March 2013
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	~~I would like to introduce accurate psychic predictions (~~[http://~~myoceancounty~~.~~net~~/~~groups/apply~~-~~these~~-~~guidelines~~-~~when~~-~~gardening~~-~~and~~-~~grow~~/ ~~myoceancounty.net~~]) ~~myself to you, I am Jayson Simcox but I don~~'~~t like when individuals use my full name. For~~ a ~~whilst I~~'~~ve been in Mississippi but now I~~'~~m contemplating other choices~~. The ~~preferred pastime for him~~ and ~~his children is fashion~~ and he'~~ll be starting some thing else along~~ [~~http~~:/~~/www~~.~~article~~-~~galaxy~~.~~com~~/~~profile.php?~~a=~~143251 free tarot readings] with it. He functions as~~ a ~~bookkeeper~~.<br><br>~~My web blog~~ ... ~~phone psychic readings~~, [~~http~~://~~clothingcarearchworth~~.~~com~~/~~index~~.~~php?document_srl~~=~~441551~~&~~mid~~=~~customer_review clothingcarearchworth~~.~~com~~],		In [[algebra]], the '''Binet–Cauchy identity''', named after [[Jacques Philippe Marie Binet]] and [[Augustin-Louis Cauchy]], states that <ref name=Weisstein>

			{{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}}

			</ref>

			: <math>
			\biggl(\sum_{i=1}^n a_i c_i\biggr)
			\biggl(\sum_{j=1}^n b_j d_j\biggr) =
			\biggl(\sum_{i=1}^n a_i d_i\biggr)
			\biggl(\sum_{j=1}^n b_j c_j\biggr)
			+ \sum_{1\le i < j \le n}
			(a_i b_j - a_j b_i )
			(c_i d_j - c_j d_i )
			</math>

			for every choice of [[real number\|real]] or [[complex number]]s (or more generally, elements of a [[commutative ring]]).
			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>.

			==The Binet–Cauchy identity and exterior algebra==
			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

			:<math>(a \cdot c)(b \cdot d) = (a \cdot d)(b \cdot c) + (a \wedge b) \cdot (c \wedge d)\,</math>

			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

			:<math>(a \wedge b) \cdot (c \wedge d) = (a \cdot c)(b \cdot d) - (a \cdot d)(b \cdot c).\,</math>
			In the special case of unit vectors ''a=c'' and ''b=d'', the formula yields
			:<math>\|a \wedge b\|^2 = \|a\|^2\|b\|^2 - \|a \cdot b\|^2. \,</math>

			When both vectors are unit vectors, we obtain the usual relation
			:<math>1= \cos^2(\phi)+\sin^2(\phi)</math>
			where φ is the angle between the vectors.

			==Proof==
			Expanding the last term,

			:<math>
			\sum_{1\le i < j \le n}
			(a_i b_j - a_j b_i )
			(c_i d_j - c_j d_i )
			</math>
			:<math>
			=
			\sum_{1\le i < j \le n}
			(a_i c_i b_j d_j + a_j c_j b_i d_i)
			+\sum_{i=1}^n a_i c_i b_i d_i
			-
			\sum_{1\le i < j \le n}
			(a_i d_i b_j c_j + a_j d_j b_i c_i)
			-
			\sum_{i=1}^n a_i d_i b_i c_i
			</math>

			where the second and fourth terms are the same and artificially added to complete the sums as follows:

			:<math>
			=
			\sum_{i=1}^n \sum_{j=1}^n
			a_i c_i b_j d_j
			-
			\sum_{i=1}^n \sum_{j=1}^n
			a_i d_i b_j c_j.
			</math>

			This completes the proof after factoring out the terms indexed by ''i''.

			==Generalization==
			A general form, also known as the [[Cauchy–Binet formula]], states the following:
			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''.
			Then the [[determinant]] of the [[matrix product]] of ''A'' and ''B'' satisfies the identity
			:<math>\det(AB) = \sum_{\scriptstyle S\subset\{1,\ldots,n\}\atop\scriptstyle\|S\|=m} \det(A_S)\det(B_S),</math>
			where the sum extends over all possible subsets ''S'' of {1, ..., ''n''} with ''m'' elements.

			We get the original identity as special case by setting
			:<math>
			A=\begin{pmatrix}a_1&\dots&a_n\\b_1&\dots& b_n\end{pmatrix},\quad
			B=\begin{pmatrix}c_1&d_1\\\vdots&\vdots\\c_n&d_n\end{pmatrix}.
			</math>

			==In-line notes and references==
			<references/>

			{{DEFAULTSORT:Binet-Cauchy Identity}}
			[[Category:Mathematical identities]]
			[[Category:Multilinear algebra]]
			[[Category:Articles containing proofs]]

en>Nageh: categories

2012-04-13T15:17:20Z