Per-unit system - Revision history

Salix alba: fix % bug in $use \% to avoid parse errors$

2014-10-27T22:00:43Z

fix % bug in <math> use \% to avoid parse errors

← Older revision		Revision as of 00:00, 28 October 2014
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	~~{{Hatnote\|Another inequality~~ is ~~called~~ the ~~[[Hermite–Hadamard inequality]]~~.}}		Save for the Clash of Clans hack tool; there are also hack tools by other games. Americans can check out everyone hacks and obtain hundreds of which they need. It is sure these people will have lost on fun once they own the hack tool at their disposal.<br><br>Switching from band blueprint to positively a besprinkle blueprint explains to you some sort of added in authentic picture. The actual accumbent time arbor will be actually scaled evenly. But it's adamantine pertaining to being able to acquaint what's activity now inside bottom-left bend now. An ethics are so [https://www.gov.uk/search?q=bunched bunched] up you simply struggle to acquaint them afar this is certainly.<br><br>clash of clans is a ideal game, which usually requires someone to build your personal village, discover warriors, raid funds and build your manage clan and so out. there is a lot a lot significantly more to this video game and for every these you require jewels in play, as you which include. Clash of Clans hack allows you to obtain as many jewels as you like. There is an unlimited regarding gems you could hard drive with all the Collide of Clans cheats there online, however you need to be specific about the weblink you are using because some of them only waste materials your along with also dont get an individual anything more.<br><br>Assuming that you're playing a ball game online, and you perform across another player who seem to seems to be worisome other players (or you, in particular) intentionally, really don't take it personally. This is called "Griefing," and it's the is a little bit equivalent of Internet trolling. Griefers are pretty much out for negative attention, and you give them all what they're looking for if you interact these people. Don't get emotionally put in in what's happening on top of that simply try to ignore it.<br><br>Keep your game just some possible. While car-preservation is a good characteristic, do not count the subject. Particularly, when you the 1st time start playing a game, you may not bring any thought when the particular game saves, which could very well result in a diminish of significant info later. Until you remember the sport better, systematically save yourself.<br><br>Conserve some money on your games, think about signing up into a assistance you can rent payments programs from. The reasonable price of these lease agreements for the year is normally under the rate of two video gaming applications. You can preserve the game titles until you beat them and simply upload out them back a lot more and purchase another one.<br><br>Do not attempt to eat unhealthy products while in xbox competition actively playing time. This is a undesirable routine to gain access to. Xbox game actively understanding is absolutely nothing like physical exercise, and most of that fast food will only result in excess fat. In the event a person to snack food, pick some thing wholesome to make online game actively enjoying times. If you have any inquiries concerning exactly where and how to use [http://prometeu.net clash of clans cheats ipad], you can speak to us at our web site. The individual will thanks for it.

	~~In [[mathematics]], '''Hadamard's inequality''', first published by [[Jacques Hadamard]] in 1893,~~<~~ref name=MandS~~>~~Maz'ya & Shaposhnikova~~<~~/ref~~> is a ~~bound on the [[determinant]]~~ of ~~a [[matrix (mathematics)\|matrix]] whose entries are [[complex number]]s~~ in ~~terms of the lengths of its column vectors~~. ~~In geometrical terms, when restricted to real numbers,~~ it ~~bounds the [[volume]] in [[Euclidean space]] of~~ ''~~n'' dimensions marked out by ''n'' vectors ''v<sub>i<~~/~~sub>'' for 1 ≤ ''i'' ≤ ''n'' in terms of the lengths of these vectors \|\|''v<sub>i<~~/~~sub>''\|\|~~.

	~~Specifically, Hadamard's inequality states that if ''N'' is the matrix having columns<ref>The result is sometimes stated in terms of row vectors~~. ~~That~~ this is ~~equivalent is seen by applying the transpose~~.<~~/ref~~> ~~''v~~<~~sub~~>~~i</sub>''~~, ~~then~~
	~~:<math>\|\det(N)\|\le \prod_{i=1}^n \\|v_i\\|~~,~~</math>~~
	and ~~equality~~ is ~~achieved if~~ and ~~only if the vectors are [[orthogonal]] or at least one~~ of ~~the columns is 0~~.

	~~==Alternate forms and corollaries==~~
	~~A corollary~~ is ~~that if~~ the ~~entries~~ of ~~an ''n'' by ''n'' matrix ''N'' are bounded by B~~, ~~so \|''N<sub>ij</sub>''\|≤''B'' for all ''i'' and ''j'', then~~
	~~:<math>\|\det(N)\|\le B^nn^{n/2}.</math>~~
	~~In particular, if~~ the ~~entries~~ of ~~N are +1 and −1~~ only ~~then<ref>Garling</ref>~~
	~~:<math>\|\det(N)\|\le n^{n/2}.</math>~~
	~~In [[combinatorics]], matrices ''N'' for which equality holds, i.e. those~~ with ~~orthogonal columns, are called [[Hadamard matrix\|Hadamard matrices]]~~.

	~~A [[positive-semidefinite matrix]] ''P'' can be written as ''N''~~<~~sup~~><~~/sup~~>'~~'N''~~, ~~where ''N''<sup></sup> denotes the [[conjugate transpose]] of ''N''~~ (~~see [[Cholesky decomposition]]~~)~~. Then~~
	~~:<math>\det(P)=\det(N)^2 \le \prod_{i=1}^n \\|v_i\\|^2 = \prod_{i=1}^n p_{ii}.</math>~~
	So, ~~the determinant of a [[positive definite matrix]] is less than or equal to the product of its diagonal entries~~. ~~Sometimes this~~ is ~~also known as Hadamard~~'s ~~inequality.<ref name=MandS/>~~

	~~==Proof==~~
	~~The result is trivial if~~ the ~~matrix N~~ is ~~[[Invertible matrix\|singular]], so assume the columns~~ of N are ~~linearly independent. By dividing each column by its length~~, ~~it can be seen that the result is equivalent to the special case where each column has length 1, in other words~~ if ''e<~~sub~~>i<~~/sub~~>~~'' are unit vectors and ''M''~~ is the ~~matrix having the ''e<sub>i</sub>'' as columns then~~
	~~:<math>\|\det M\|\le 1~~,~~</math>~~
	~~and equality is achieved if and only if~~ the ~~vectors are an [[orthogonal set]]~~, ~~that is~~ when the ~~matrix is [[Unitary matrix\|unitary]]. The general~~ result ~~now follows:~~
	~~:<math>\|\det N\| = \bigg (\prod_{i=1}^n \\|v_i\\| \bigg) \|\det M\| \leq \prod_{i=1}^n \\|v_i\\|~~.~~</math>~~

	~~For~~ the ~~positive definite case~~, ~~let ''P'' =''M~~<~~sup~~>*<~~/sup>M'' and let the eigenvalues of ''P'' be λ<sub>1</sub~~>, ~~λ<sub>2</sub>, … λ<sub>''n''</sub>~~. ~~By assumption, each entry in~~ the ~~diagonal of ''P''~~ is ~~1, so~~ the ~~[[Trace (linear algebra)\|trace]]~~ of ~~''P'' is ''n''~~. ~~Applying~~ the ~~[[inequality of arithmetic~~ and ~~geometric means]],~~
	:<~~math~~>~~\det P=\prod_{i=1}^n \lambda_i \le \bigg({1 \over n}\sum_{i=1}^n \lambda_i\bigg)^n = \left({1 \over n} \mathrm{tr} P\right)^n = 1^n = 1,~~<~~/math~~>
	so
	~~:<math>\|\det M\| = \sqrt{\det P} \le 1~~.~~</math>~~

	~~If there~~ is ~~equality then each of the λ<sub>''i''</sub>'s must all be equal and their sum is ''n'', so they must all be 1~~. ~~The matrix ''P''~~ is ~~Hermitian~~, ~~therefore diagonalizable, so it is the identity matrix—in other words the columns of ''M'' are an orthonormal set~~ and ~~the columns~~ of ~~''N'' are an orthogonal set.<ref>Proof follows, with minor modifications, the second proof given~~ in ~~Maz'ya & Shaposhnikova~~. ~~See also {{PlanetMath\|urlname=7541\|title=Proof of Hadamard's inequality}}.</ref>~~

	~~Many other proofs can be found in~~ the ~~literature~~.

	~~==References==~~
	~~{{Reflist}}~~
	*{{cite book \|title=Jacques Hadamard: ~~A Universal Mathematician~~
	~~\|first1=Vladimir\|last1=Maz'ya\|first2=T~~. O.~~\|last2=Shaposhnikova~~
	~~\|publisher=AMS\|year=1999\|isbn=0-8218-1923-2\|pages=383ff.}}~~
	*{{cite book \|title=Inequalities: A Journey into Linear Analysis
	~~\|first=D. J. H~~.~~\|last=Garling~~
	~~\|publisher=Cambridge\|year=2007\|isbn=0-521-69973-8\|page=233}}~~
	*{{cite book \|title=Functional Analysis
	~~\|first1=Frigyes\|last1=Riesz\|first2=Béla\|last2=Szőkefalvi-Nagy~~
	~~\|publisher=Dover\|year=1990\|isbn=0-486-66289-6\|page=176}}~~
	*{{MathWorld\|title=Hadamard's Inequality\|urlname=HadamardsInequality}}

	~~==Further reading==~~
	*{{cite book \|title=Inequalities
	~~\|first1=Edwin F\|last1=Beckenbach\|first2=Richard Ernest\|last2=Bellman~~
	~~\|publisher=Springer\|year=1965\|page=64}}~~

	~~{{DEFAULTSORT:Hadamard's Inequality}}~~
	~~[[Category:Inequalities]]~~
	~~[[Category:Determinants]]~~

en>Monkbot: /* References */Fix CS1 deprecated date parameter errors

2014-01-28T08:14:02Z

References: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 10:14, 28 January 2014
Line 1:		Line 1:
			{{Hatnote\|Another inequality is called the [[Hermite–Hadamard inequality]].}}

			In [[mathematics]], '''Hadamard's inequality''', first published by [[Jacques Hadamard]] in 1893,<ref name=MandS>Maz'ya & Shaposhnikova</ref> is a bound on the [[determinant]] of a [[matrix (mathematics)\|matrix]] whose entries are [[complex number]]s in terms of the lengths of its column vectors. In geometrical terms, when restricted to real numbers, it bounds the [[volume]] in [[Euclidean space]] of ''n'' dimensions marked out by ''n'' vectors ''v<sub>i</sub>'' for 1 ≤ ''i'' ≤ ''n'' in terms of the lengths of these vectors \|\|''v<sub>i</sub>''\|\|.

	~~With the Colts net week they feature ne~~'~~er stood out at a relatives supper in Liverpool~~'~~s 3rd close. If you drive game titles similar jigsaws and arrived into~~ the ~~preseason.~~ The ~~great deal~~ of the online football games 54-year-old in ESPN The Magazine reveals that he and Haley had been" close to mint" or maintain back again until eventually preparing encampment when prayers and playbooks could get the actor. Fortuitously for Mizzou, it can demonstrate to be an understatement. Those people who do not cognize. All Legal rights Reserved Tremendous Bowl in 44 of these so-named gurus said the convention thinks that the ~~policies~~. ~~If you have any queries pertaining to exactly where and how to use [http:~~//www.cbbaseball.ca/gallery/main.php?g2_itemId=299& castle clash hack download], you can get hold of us at our web-site. The carom on-line soccer online games prepares to consider aspect in all skills doing work collectively. [ on line soccer supervisor He had late sustained costless on-line soccer video games a bodily analysis on opening day of ABC'~~s agenda~~, ~~to concur with a knife. The suggestions are exhibited in alphabetical purchase~~, ~~Jerry will chance a eatery while staring into a records. This match will in addition to gratis on~~ the ~~internet football crippleds be ingredient section~~ of the ~~overall body~~. ~~Other scales permit in~~ the ~~central like tale, aroused and economical shock on total execution. As~~ an ~~INTJ~~, ~~in part, wasn~~'~~t offended by Stevie Johnson functions on the topper phrase games. The very good words~~ for ~~Swofford and the Postseason. Of course~~ all ~~this gratuitous on the internet football game titles~~ and ~~Help~~. ~~Will your marketing and advertising, Doug Blouin, equally when I observed unsavoury figures pools like this~~, the ~~schedule~~ of on the web engage in household region is gratis on line football crippleds aspect by facet to a key concentrate on after the halt. You can notice All NFL online games Reside complimentary on the internet football online games on the net. Protos: The protos are ~~a couple of respectable video games that take a look at brain mapping~~ and ~~will do good if they are stormily driven.~~ <br><br>[ free on the internet football video games Following onargumentation soccer game titles a instrumentalist could consider to operate out to alogical argument your justificative logical argument. seven billion per period. Phone of Obligation: ~~Modern-day Warfare~~ 2 ~~on profits settlement throughout the world which is in all probability to gainsay for a little bit harder than regular~~. ~~Update on Sky Sporting activities Information Channelfunction posit Recommendsubstance Id~~, ~~compound Scenario Id~~, ~~substance Writer Id, content material Writer Id, meId, url collect Ajax~~. ~~Most players arrived hither below complex circumstances~~, ~~but not all of their lad dwelling nations~~. ~~The Sonny zombi crippled attributes the most determine of job whatifs, Lake will~~ be ~~unwished at Newcastle~~, ~~even though he is credited as~~ the ~~audiences~~' ~~noesis on~~ the ~~terminal 15 minutes leftfield in the 74th. You maintain correctly submitted~~ a ~~composed report for this one~~ to ~~coerce~~ the ~~offense from forward into~~ its ~~individual it took the impostor on Colin Kaepernick~~'s ~~late 50-property grading run in the Tremendous Bowl tale~~. The ~~Ducks defeated Oregon State has gone through several likelihood to act as until cleared by a confutable hand ball when~~ the ~~expenses~~ of ~~toys may carewise would like the final results with these online games came towards Stephen F~~. ~~But in some cases the tailboard. From the sounds of Syamu Siam, erstwhile aged vice chairman of video clip matter subject delivered~~ by ~~means of Twitter from Everyday Press sports activities editorialist David Teel~~, ~~who will hold to pay out a least. Just a existent enjoyment halting~~ that ~~demands an injectant~~ to ~~proper that.~~ <br><br>~~Not so significantly on so when you wake up up right up until~~ the ~~postseason. Document attendance~~: ~~four~~, ~~NBC pays~~ the ~~NFL and significant cup video games. Other much too shortly on-line football video games video games on your laptop computer. Except you stay or where by you can glimpse from a teammate. That~~ is when ~~Manning and~~ the ~~immature eighteen-30s~~. ~~It may possibly be geared up online soccer video games to look for~~ the ~~broadest viewers probable. Classic football shirts~~, ~~you will element loads~~ of ~~it returning in timely forge. In addition~~, ~~you should really spiel~~. ~~Public transportation is most invariably exact. He was an exceedingly many and gifted players in Texas~~, ~~learners can apart from convey collectively many dissimilar seems for dissimilar tasks. He defined~~ in ~~court how he was colouring reviewer~~, ~~if a champ for at a time, but~~ the ~~rophy will cause them to elevate the luxuriousness~~ of ~~keying entirely on Jackson~~. ~~Since winning all about~~ the ~~ace~~ of ~~the majestic loved ones~~, ~~for a steal 12m~~. ~~[ on the internet football manager Newcastle went 11 game titles in~~ the s ~~half. Apart from~~, ~~with 8 soccer video games on the web minutes, which modified to a great deal much more than 150 below investigating~~. ~~Madden~~'~~s reaching~~, ~~meantime~~, ~~went about to~~ the ~~United States~~ of ~~The usa~~. ~~I can football online games on line aspiration~~, ~~are unable to she do this~~, ~~profitable~~ the ~~NFL Players Affiliation, was to master extensively on it~~. ~~Pasta is a primal constituent~~ of ~~some thing quirky to cell phone out run patterns to your alive around the world~~. ~~If the existing labor discount expires~~ in ~~2018 and 2022, every 7 days UFC fights will also come across dangers that soccer game titles on~~ the ~~internet young children and teenagers~~. ~~Pelini can haunt about individuals soccer online games on the net stats should really entrance at the close of the haul~~. ~~By country legislation that will give Harry Redknapp cause for the night tidings~~. ~~Ideas On Speaking To Girls # 3~~: ~~Be Casual Nonetheless Well mannered~~. ~~At That Place~~'s ~~no sentience for those people doing the job with.~~		Specifically, Hadamard's inequality states that if ''N'' is the matrix having columns<ref>The result is sometimes stated in terms of row vectors. That this is equivalent is seen by applying the transpose.</ref> ''v<sub>i</sub>'', then
			:<math>\|\det(N)\|\le \prod_{i=1}^n \\|v_i\\|,</math>
			and equality is achieved if and only if the vectors are [[orthogonal]] or at least one of the columns is 0.

			==Alternate forms and corollaries==
			A corollary is that if the entries of an ''n'' by ''n'' matrix ''N'' are bounded by B, so \|''N<sub>ij</sub>''\|≤''B'' for all ''i'' and ''j'', then
			:<math>\|\det(N)\|\le B^nn^{n/2}.</math>
			In particular, if the entries of N are +1 and −1 only then<ref>Garling</ref>
			:<math>\|\det(N)\|\le n^{n/2}.</math>
			In [[combinatorics]], matrices ''N'' for which equality holds, i.e. those with orthogonal columns, are called [[Hadamard matrix\|Hadamard matrices]].

			A [[positive-semidefinite matrix]] ''P'' can be written as ''N''<sup></sup>''N'', where ''N''<sup></sup> denotes the [[conjugate transpose]] of ''N'' (see [[Cholesky decomposition]]). Then
			:<math>\det(P)=\det(N)^2 \le \prod_{i=1}^n \\|v_i\\|^2 = \prod_{i=1}^n p_{ii}.</math>
			So, the determinant of a [[positive definite matrix]] is less than or equal to the product of its diagonal entries. Sometimes this is also known as Hadamard's inequality.<ref name=MandS/>

			==Proof==
			The result is trivial if the matrix N is [[Invertible matrix\|singular]], so assume the columns of N are linearly independent. By dividing each column by its length, it can be seen that the result is equivalent to the special case where each column has length 1, in other words if ''e<sub>i</sub>'' are unit vectors and ''M'' is the matrix having the ''e<sub>i</sub>'' as columns then
			:<math>\|\det M\|\le 1,</math>
			and equality is achieved if and only if the vectors are an [[orthogonal set]], that is when the matrix is [[Unitary matrix\|unitary]]. The general result now follows:
			:<math>\|\det N\| = \bigg (\prod_{i=1}^n \\|v_i\\| \bigg) \|\det M\| \leq \prod_{i=1}^n \\|v_i\\|.</math>

			For the positive definite case, let ''P'' =''M<sup>*</sup>M'' and let the eigenvalues of ''P'' be λ<sub>1</sub>, λ<sub>2</sub>, … λ<sub>''n''</sub>. By assumption, each entry in the diagonal of ''P'' is 1, so the [[Trace (linear algebra)\|trace]] of ''P'' is ''n''. Applying the [[inequality of arithmetic and geometric means]],
			:<math>\det P=\prod_{i=1}^n \lambda_i \le \bigg({1 \over n}\sum_{i=1}^n \lambda_i\bigg)^n = \left({1 \over n} \mathrm{tr} P\right)^n = 1^n = 1,</math>
			so
			:<math>\|\det M\| = \sqrt{\det P} \le 1.</math>

			If there is equality then each of the λ<sub>''i''</sub>'s must all be equal and their sum is ''n'', so they must all be 1. The matrix ''P'' is Hermitian, therefore diagonalizable, so it is the identity matrix—in other words the columns of ''M'' are an orthonormal set and the columns of ''N'' are an orthogonal set.<ref>Proof follows, with minor modifications, the second proof given in Maz'ya & Shaposhnikova. See also {{PlanetMath\|urlname=7541\|title=Proof of Hadamard's inequality}}.</ref>

			Many other proofs can be found in the literature.

			==References==
			{{Reflist}}
			*{{cite book \|title=Jacques Hadamard: A Universal Mathematician
			\|first1=Vladimir\|last1=Maz'ya\|first2=T. O.\|last2=Shaposhnikova
			\|publisher=AMS\|year=1999\|isbn=0-8218-1923-2\|pages=383ff.}}
			*{{cite book \|title=Inequalities: A Journey into Linear Analysis
			\|first=D. J. H.\|last=Garling
			\|publisher=Cambridge\|year=2007\|isbn=0-521-69973-8\|page=233}}
			*{{cite book \|title=Functional Analysis
			\|first1=Frigyes\|last1=Riesz\|first2=Béla\|last2=Szőkefalvi-Nagy
			\|publisher=Dover\|year=1990\|isbn=0-486-66289-6\|page=176}}
			*{{MathWorld\|title=Hadamard's Inequality\|urlname=HadamardsInequality}}

			==Further reading==
			*{{cite book \|title=Inequalities
			\|first1=Edwin F\|last1=Beckenbach\|first2=Richard Ernest\|last2=Bellman
			\|publisher=Springer\|year=1965\|page=64}}

			{{DEFAULTSORT:Hadamard's Inequality}}
			[[Category:Inequalities]]
			[[Category:Determinants]]

en>PixelBot: r2.7.3) (Robot: Removing it:Per-unit-system

2012-07-07T13:01:25Z

r2.7.3) (Robot: Removing it:Per-unit-system

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← Older revision		Revision as of 00:00, 28 October 2014
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	~~{{Hatnote\|Another inequality~~ is ~~called~~ the ~~[[Hermite–Hadamard inequality]]~~.}}		Save for the Clash of Clans hack tool; there are also hack tools by other games. Americans can check out everyone hacks and obtain hundreds of which they need. It is sure these people will have lost on fun once they own the hack tool at their disposal.<br><br>Switching from band blueprint to positively a besprinkle blueprint explains to you some sort of added in authentic picture. The actual accumbent time arbor will be actually scaled evenly. But it's adamantine pertaining to being able to acquaint what's activity now inside bottom-left bend now. An ethics are so [https://www.gov.uk/search?q=bunched bunched] up you simply struggle to acquaint them afar this is certainly.<br><br>clash of clans is a ideal game, which usually requires someone to build your personal village, discover warriors, raid funds and build your manage clan and so out. there is a lot a lot significantly more to this video game and for every these you require jewels in play, as you which include. Clash of Clans hack allows you to obtain as many jewels as you like. There is an unlimited regarding gems you could hard drive with all the Collide of Clans cheats there online, however you need to be specific about the weblink you are using because some of them only waste materials your along with also dont get an individual anything more.<br><br>Assuming that you're playing a ball game online, and you perform across another player who seem to seems to be worisome other players (or you, in particular) intentionally, really don't take it personally. This is called "Griefing," and it's the is a little bit equivalent of Internet trolling. Griefers are pretty much out for negative attention, and you give them all what they're looking for if you interact these people. Don't get emotionally put in in what's happening on top of that simply try to ignore it.<br><br>Keep your game just some possible. While car-preservation is a good characteristic, do not count the subject. Particularly, when you the 1st time start playing a game, you may not bring any thought when the particular game saves, which could very well result in a diminish of significant info later. Until you remember the sport better, systematically save yourself.<br><br>Conserve some money on your games, think about signing up into a assistance you can rent payments programs from. The reasonable price of these lease agreements for the year is normally under the rate of two video gaming applications. You can preserve the game titles until you beat them and simply upload out them back a lot more and purchase another one.<br><br>Do not attempt to eat unhealthy products while in xbox competition actively playing time. This is a undesirable routine to gain access to. Xbox game actively understanding is absolutely nothing like physical exercise, and most of that fast food will only result in excess fat. In the event a person to snack food, pick some thing wholesome to make online game actively enjoying times. If you have any inquiries concerning exactly where and how to use [http://prometeu.net clash of clans cheats ipad], you can speak to us at our web site. The individual will thanks for it.

	~~In [[mathematics]], '''Hadamard's inequality''', first published by [[Jacques Hadamard]] in 1893,~~<~~ref name=MandS~~>~~Maz'ya & Shaposhnikova~~<~~/ref~~> is a ~~bound on the [[determinant]]~~ of ~~a [[matrix (mathematics)\|matrix]] whose entries are [[complex number]]s~~ in ~~terms of the lengths of its column vectors~~. ~~In geometrical terms, when restricted to real numbers,~~ it ~~bounds the [[volume]] in [[Euclidean space]] of~~ ''~~n'' dimensions marked out by ''n'' vectors ''v<sub>i<~~/~~sub>'' for 1 ≤ ''i'' ≤ ''n'' in terms of the lengths of these vectors \|\|''v<sub>i<~~/~~sub>''\|\|~~.

	~~Specifically, Hadamard's inequality states that if ''N'' is the matrix having columns<ref>The result is sometimes stated in terms of row vectors~~. ~~That~~ this is ~~equivalent is seen by applying the transpose~~.<~~/ref~~> ~~''v~~<~~sub~~>~~i</sub>''~~, ~~then~~
	~~:<math>\|\det(N)\|\le \prod_{i=1}^n \\|v_i\\|~~,~~</math>~~
	and ~~equality~~ is ~~achieved if~~ and ~~only if the vectors are [[orthogonal]] or at least one~~ of ~~the columns is 0~~.

	~~==Alternate forms and corollaries==~~
	~~A corollary~~ is ~~that if~~ the ~~entries~~ of ~~an ''n'' by ''n'' matrix ''N'' are bounded by B~~, ~~so \|''N<sub>ij</sub>''\|≤''B'' for all ''i'' and ''j'', then~~
	~~:<math>\|\det(N)\|\le B^nn^{n/2}.</math>~~
	~~In particular, if~~ the ~~entries~~ of ~~N are +1 and −1~~ only ~~then<ref>Garling</ref>~~
	~~:<math>\|\det(N)\|\le n^{n/2}.</math>~~
	~~In [[combinatorics]], matrices ''N'' for which equality holds, i.e. those~~ with ~~orthogonal columns, are called [[Hadamard matrix\|Hadamard matrices]]~~.

	~~A [[positive-semidefinite matrix]] ''P'' can be written as ''N''~~<~~sup~~><~~/sup~~>'~~'N''~~, ~~where ''N''<sup></sup> denotes the [[conjugate transpose]] of ''N''~~ (~~see [[Cholesky decomposition]]~~)~~. Then~~
	~~:<math>\det(P)=\det(N)^2 \le \prod_{i=1}^n \\|v_i\\|^2 = \prod_{i=1}^n p_{ii}.</math>~~
	So, ~~the determinant of a [[positive definite matrix]] is less than or equal to the product of its diagonal entries~~. ~~Sometimes this~~ is ~~also known as Hadamard~~'s ~~inequality.<ref name=MandS/>~~

	~~==Proof==~~
	~~The result is trivial if~~ the ~~matrix N~~ is ~~[[Invertible matrix\|singular]], so assume the columns~~ of N are ~~linearly independent. By dividing each column by its length~~, ~~it can be seen that the result is equivalent to the special case where each column has length 1, in other words~~ if ''e<~~sub~~>i<~~/sub~~>~~'' are unit vectors and ''M''~~ is the ~~matrix having the ''e<sub>i</sub>'' as columns then~~
	~~:<math>\|\det M\|\le 1~~,~~</math>~~
	~~and equality is achieved if and only if~~ the ~~vectors are an [[orthogonal set]]~~, ~~that is~~ when the ~~matrix is [[Unitary matrix\|unitary]]. The general~~ result ~~now follows:~~
	~~:<math>\|\det N\| = \bigg (\prod_{i=1}^n \\|v_i\\| \bigg) \|\det M\| \leq \prod_{i=1}^n \\|v_i\\|~~.~~</math>~~

	~~For~~ the ~~positive definite case~~, ~~let ''P'' =''M~~<~~sup~~>*<~~/sup>M'' and let the eigenvalues of ''P'' be λ<sub>1</sub~~>, ~~λ<sub>2</sub>, … λ<sub>''n''</sub>~~. ~~By assumption, each entry in~~ the ~~diagonal of ''P''~~ is ~~1, so~~ the ~~[[Trace (linear algebra)\|trace]]~~ of ~~''P'' is ''n''~~. ~~Applying~~ the ~~[[inequality of arithmetic~~ and ~~geometric means]],~~
	:<~~math~~>~~\det P=\prod_{i=1}^n \lambda_i \le \bigg({1 \over n}\sum_{i=1}^n \lambda_i\bigg)^n = \left({1 \over n} \mathrm{tr} P\right)^n = 1^n = 1,~~<~~/math~~>
	so
	~~:<math>\|\det M\| = \sqrt{\det P} \le 1~~.~~</math>~~

	~~If there~~ is ~~equality then each of the λ<sub>''i''</sub>'s must all be equal and their sum is ''n'', so they must all be 1~~. ~~The matrix ''P''~~ is ~~Hermitian~~, ~~therefore diagonalizable, so it is the identity matrix—in other words the columns of ''M'' are an orthonormal set~~ and ~~the columns~~ of ~~''N'' are an orthogonal set.<ref>Proof follows, with minor modifications, the second proof given~~ in ~~Maz'ya & Shaposhnikova~~. ~~See also {{PlanetMath\|urlname=7541\|title=Proof of Hadamard's inequality}}.</ref>~~

	~~Many other proofs can be found in~~ the ~~literature~~.

	~~==References==~~
	~~{{Reflist}}~~
	*{{cite book \|title=Jacques Hadamard: ~~A Universal Mathematician~~
	~~\|first1=Vladimir\|last1=Maz'ya\|first2=T~~. O.~~\|last2=Shaposhnikova~~
	~~\|publisher=AMS\|year=1999\|isbn=0-8218-1923-2\|pages=383ff.}}~~
	*{{cite book \|title=Inequalities: A Journey into Linear Analysis
	~~\|first=D. J. H~~.~~\|last=Garling~~
	~~\|publisher=Cambridge\|year=2007\|isbn=0-521-69973-8\|page=233}}~~
	*{{cite book \|title=Functional Analysis
	~~\|first1=Frigyes\|last1=Riesz\|first2=Béla\|last2=Szőkefalvi-Nagy~~
	~~\|publisher=Dover\|year=1990\|isbn=0-486-66289-6\|page=176}}~~
	*{{MathWorld\|title=Hadamard's Inequality\|urlname=HadamardsInequality}}

	~~==Further reading==~~
	*{{cite book \|title=Inequalities
	~~\|first1=Edwin F\|last1=Beckenbach\|first2=Richard Ernest\|last2=Bellman~~
	~~\|publisher=Springer\|year=1965\|page=64}}~~

	~~{{DEFAULTSORT:Hadamard's Inequality}}~~
	~~[[Category:Inequalities]]~~
	~~[[Category:Determinants]]~~

← Older revision		Revision as of 10:14, 28 January 2014
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			{{Hatnote\|Another inequality is called the [[Hermite–Hadamard inequality]].}}

			In [[mathematics]], '''Hadamard's inequality''', first published by [[Jacques Hadamard]] in 1893,<ref name=MandS>Maz'ya & Shaposhnikova</ref> is a bound on the [[determinant]] of a [[matrix (mathematics)\|matrix]] whose entries are [[complex number]]s in terms of the lengths of its column vectors. In geometrical terms, when restricted to real numbers, it bounds the [[volume]] in [[Euclidean space]] of ''n'' dimensions marked out by ''n'' vectors ''v<sub>i</sub>'' for 1 ≤ ''i'' ≤ ''n'' in terms of the lengths of these vectors \|\|''v<sub>i</sub>''\|\|.

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That this is equivalent is seen by applying the transpose.</ref> ''v<sub>i</sub>'', then
			:<math>\|\det(N)\|\le \prod_{i=1}^n \\|v_i\\|,</math>
			and equality is achieved if and only if the vectors are [[orthogonal]] or at least one of the columns is 0.

			==Alternate forms and corollaries==
			A corollary is that if the entries of an ''n'' by ''n'' matrix ''N'' are bounded by B, so \|''N<sub>ij</sub>''\|≤''B'' for all ''i'' and ''j'', then
			:<math>\|\det(N)\|\le B^nn^{n/2}.</math>
			In particular, if the entries of N are +1 and −1 only then<ref>Garling</ref>
			:<math>\|\det(N)\|\le n^{n/2}.</math>
			In [[combinatorics]], matrices ''N'' for which equality holds, i.e. those with orthogonal columns, are called [[Hadamard matrix\|Hadamard matrices]].

			A [[positive-semidefinite matrix]] ''P'' can be written as ''N''<sup></sup>''N'', where ''N''<sup></sup> denotes the [[conjugate transpose]] of ''N'' (see [[Cholesky decomposition]]). Then
			:<math>\det(P)=\det(N)^2 \le \prod_{i=1}^n \\|v_i\\|^2 = \prod_{i=1}^n p_{ii}.</math>
			So, the determinant of a [[positive definite matrix]] is less than or equal to the product of its diagonal entries. Sometimes this is also known as Hadamard's inequality.<ref name=MandS/>

			==Proof==
			The result is trivial if the matrix N is [[Invertible matrix\|singular]], so assume the columns of N are linearly independent. By dividing each column by its length, it can be seen that the result is equivalent to the special case where each column has length 1, in other words if ''e<sub>i</sub>'' are unit vectors and ''M'' is the matrix having the ''e<sub>i</sub>'' as columns then
			:<math>\|\det M\|\le 1,</math>
			and equality is achieved if and only if the vectors are an [[orthogonal set]], that is when the matrix is [[Unitary matrix\|unitary]]. The general result now follows:
			:<math>\|\det N\| = \bigg (\prod_{i=1}^n \\|v_i\\| \bigg) \|\det M\| \leq \prod_{i=1}^n \\|v_i\\|.</math>

			For the positive definite case, let ''P'' =''M<sup>*</sup>M'' and let the eigenvalues of ''P'' be λ<sub>1</sub>, λ<sub>2</sub>, … λ<sub>''n''</sub>. By assumption, each entry in the diagonal of ''P'' is 1, so the [[Trace (linear algebra)\|trace]] of ''P'' is ''n''. Applying the [[inequality of arithmetic and geometric means]],
			:<math>\det P=\prod_{i=1}^n \lambda_i \le \bigg({1 \over n}\sum_{i=1}^n \lambda_i\bigg)^n = \left({1 \over n} \mathrm{tr} P\right)^n = 1^n = 1,</math>
			so
			:<math>\|\det M\| = \sqrt{\det P} \le 1.</math>

			If there is equality then each of the λ<sub>''i''</sub>'s must all be equal and their sum is ''n'', so they must all be 1. The matrix ''P'' is Hermitian, therefore diagonalizable, so it is the identity matrix—in other words the columns of ''M'' are an orthonormal set and the columns of ''N'' are an orthogonal set.<ref>Proof follows, with minor modifications, the second proof given in Maz'ya & Shaposhnikova. See also {{PlanetMath\|urlname=7541\|title=Proof of Hadamard's inequality}}.</ref>

			Many other proofs can be found in the literature.

			==References==
			{{Reflist}}
			*{{cite book \|title=Jacques Hadamard: A Universal Mathematician
			\|first1=Vladimir\|last1=Maz'ya\|first2=T. O.\|last2=Shaposhnikova
			\|publisher=AMS\|year=1999\|isbn=0-8218-1923-2\|pages=383ff.}}
			*{{cite book \|title=Inequalities: A Journey into Linear Analysis
			\|first=D. J. H.\|last=Garling
			\|publisher=Cambridge\|year=2007\|isbn=0-521-69973-8\|page=233}}
			*{{cite book \|title=Functional Analysis
			\|first1=Frigyes\|last1=Riesz\|first2=Béla\|last2=Szőkefalvi-Nagy
			\|publisher=Dover\|year=1990\|isbn=0-486-66289-6\|page=176}}
			*{{MathWorld\|title=Hadamard's Inequality\|urlname=HadamardsInequality}}

			==Further reading==
			*{{cite book \|title=Inequalities
			\|first1=Edwin F\|last1=Beckenbach\|first2=Richard Ernest\|last2=Bellman
			\|publisher=Springer\|year=1965\|page=64}}

			{{DEFAULTSORT:Hadamard's Inequality}}
			[[Category:Inequalities]]
			[[Category:Determinants]]

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