Optimization problem - Revision history

87.24.210.163 at 15:31, 30 November 2014

2014-11-30T15:31:06Z

← Older revision		Revision as of 17:31, 30 November 2014
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173.212.158.59: NP-hard problems relate to optimization problems, NP-complete problems relate to decision problems. Minor fix.

2014-02-12T00:30:54Z

NP-hard problems relate to optimization problems, NP-complete problems relate to decision problems. Minor fix.

← Older revision		Revision as of 02:30, 12 February 2014
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	~~'''Sylvester's law of inertia'''~~ is ~~a [[theorem]] in [[matrix algebra]] about certain properties of~~ the ~~[[coefficient matrix]] of a [[real number\|real]] [[quadratic form]] that remain [[invariant (mathematics)\|invariant]] under a change of [[coordinate]]s. Namely, if~~ ''A'' is the [[symmetric matrix]] that defines the quadratic form, and ''S'' is any invertible matrix such that ''D'' = ''SAS''<sup>T</sup> is diagonal, then the number of negative elements in the diagonal of ''D'' is always the same, for all such ''S''; and the same goes for the number of positive elements.		Nestor is the name my parents gave me but I don't like when individuals use my complete title. His working day occupation is a financial officer but he plans on altering it. I currently live in Alabama. Bottle tops collecting is the only hobby his spouse doesn't approve of.<br><br>Review my blog: [http://nationalenergy.tt/ActivityFeed/MyProfile/tabid/60/userId/15827/Default.aspx http://nationalenergy.tt/]

	~~This property~~ is ~~named after [[J. J~~. ~~Sylvester]] who published its proof~~ in ~~1852~~.~~<ref name=syl852>~~
	~~{{cite journal\|author=Sylvester, J J \| title=A demonstration of the theorem that every homogeneous quadratic polynomial~~ is reducible by real orthogonal substitutions to the form of a sum of positive and negative squares \| journal=Philosophical Magazine (Ser. 4)\| volume=4 \| issue=23 \| pages=138–142 \| year=1852 \| url=http://www.maths.ed.ac.uk/~aar/sylv/inertia.pdf \| doi= 10.1080/14786445208647087 \| accessdate=2008-06-27}}
	~~</ref><ref name=norm>~~
	~~{{cite book\|author=Norman, C.W.\| title=Undergraduate algebra \| publisher=[[Oxford University Press]] \| pages=360–361 \| year=1986 \| isbn=0-19-853248-2 }}~~
	~~</ref>~~

	~~== Statement of the theorem ==~~
	~~Let ''A'' be a symmetric square matrix of order ''n'' with [[real number\|real]] entries. Any [[non-singular matrix]] ''S'' of~~ the same size is said to transform ''A'' into another symmetric matrix {{nowrap\|1=''B'' = ''SAS''<sup>T</sup>}}, also of order ''n'', where ''S''<sup>T</sup> is the transpose of ''S''. If ''A'' is the coefficient matrix of some quadratic form of '''R'''<sup>''n''</sup>, then ''B'' is the matrix for the same form after the change of coordinates defined by ''S''.

	~~A symmetric matrix ''A'' can always be transformed in this way into a [[diagonal matrix]] ''D'' which has~~ only ~~entries 0, +1 and −1 along the diagonal. Sylvester~~'~~s law of inertia states that the number of diagonal entries~~ of ~~each kind is an invariant of ''A'', i~~.~~e. it does not depend on the matrix ''S'' used.~~

	~~The number of +1s, denoted ''n''~~<~~sub~~>+<~~/sub~~>, is called the '''positive index of inertia''' of ''A'', and the number of −1s, denoted ''n''<sub>−</sub>, is called the '''negative index of inertia'''. The number of 0s, denoted ''n''<sub>0</sub>, is the dimension of the [~~[kernel (linear algebra)\|kernel]] of ''A'', and also the corank of ''A''. These numbers satisfy an obvious relation~~

	: ~~<math> n_0+n_{+}+n_{-}=n.\ <~~/~~math>~~

	~~The difference sign(''A'') = ''n''<sub>−<~~/~~sub> − ''n''<sub>+</sub> is usually called the '''signature''' of ''A''~~. ~~(However, some authors use that term for the whole triple {{nowrap\|(''n''<sub>0<~~/~~sub>, ''n''<sub>+</sub>, ''n''<sub>−<~~/sub>)}} consisting of the corank and the positive and negative indices of inertia of ''A''; for a non-degenerate form of a given dimension these are equivalent data, but in general the triple yields more data)

	If the matrix ''A'' has the property that every principal upper left {{nowrap\|''k'' × ''k''}} [[minor (determinant)\|minor]] ''Δ''<sub>''k''</sub> is non-zero then the negative index of inertia is equal to the number of sign changes in the sequence

	~~: <math> \Delta_0=1, \Delta_1, \ldots, \Delta_n=\det A. </math>~~

	~~==Statement in terms of eigenvalues==~~
	The positive and negative indices of a symmetric matrix ''A'' are also the number of positive and negative [[eigenvalue]]s of ''A''. Any symmetric real matrix ''A'' has an [[eigendecomposition]] of the form ''QEQ''<sup>T</sup> where ''E'' is a diagonal matrix containing the eigenvalues of ''A'', and ''Q'' is an [[orthonormal]] square matrix containing the eigenvectors. The matrix ''E'' can be written ''E'' = ''WDW''<sup>T</~~sup> where ''D'' is diagonal with entries 0, +1, or −1, and ''W'' is diagonal with ''W''<sub>''ii''<~~/~~sub> = √\|''E''<sub>''ii''<~~/~~sub>\|. The matrix ''S'' = ''QW'' transforms ''D'' to ''A''.~~

	~~== Law of inertia for quadratic forms ==~~
	In the context of [[quadratic form]]s, a real quadratic form ''Q'' in ''n'' variables (or on an ''n''-dimensional real vector space) can by a suitable change of basis (by non-singular linear transformation from x to y) be brought to the diagonal form

	~~: <math> Q(x_1,x_2,\ldots,x_n)=\sum_{i=1}^n a_i x_i^2 <~~/~~math>~~

	~~with each ''a''<sub>''i''<~~/~~sub> ∈ {0, 1, −1}~~. Sylvester's law of inertia states that the number of coefficients of a given sign is an invariant of ''Q'', i.e. does not depend on a particular choice of diagonalizing basis. Expressed geometrically, the law of inertia says that all maximal subspaces on which the restriction of the quadratic form is [[definite bilinear form\|positive definite]] (respectively, negative definite) have the same [[dimension (linear algebra)\|dimension]]. These dimensions are the positive and negative indices of inertia.

	~~==See also==~~
	*[[Metric signature]]
	*[[Morse theory]]
	*[[Cholesky decomposition]]

	~~==References==~~
	~~{{reflist}}~~
	* {{cite book \| last=Garling \| first=D. J. H. \| title=Clifford algebras. An introduction \| series=London Mathematical Society Student Texts \| volume=78 \| location=Cambridge \| publisher=[[Cambridge University Press]] \| year=2011 \| isbn=978-1-107-09638-7 \| zbl=1235.15025 }}

	~~==External links==~~
	*[http://~~planetmath~~.~~org~~/~~encyclopedia/SylvestersLaw.html Sylvester's law] on [[PlanetMath]].~~

	~~[[Category:Linear algebra]]~~
	~~[[Category:Matrix theory]]~~
	~~[[Category:Quadratic forms]]~~
	~~[[Category:Theorems in algebra]~~]

85.243.106.174: /* Continuous optimization problem */

2013-12-18T00:09:18Z

Continuous optimization problem

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en>Isheden: Undid revision 508754516 by 197.28.117.67 (talk) No such word

2012-08-24T13:12:37Z

Undid revision 508754516 by 197.28.117.67 (talk) No such word

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