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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''&nbsp;=&nbsp;''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.
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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 &minus;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''&nbsp;=&nbsp;''WDW''<sup>T</sup> where ''D'' is diagonal with entries&nbsp;0,&nbsp;+1, or&nbsp;&minus;1, and ''W'' is diagonal with ''W''<sub>''ii''</sub>&nbsp;=&nbsp;√|''E''<sub>''ii''</sub>|. The matrix ''S''&nbsp;=&nbsp;''QW'' transforms ''D'' to&nbsp;''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>&nbsp;∈&nbsp;{0,&nbsp;1,&nbsp;&minus;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]]

Latest revision as of 16:31, 30 November 2014

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