Sylvester's law of inertia

Summary

Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis. Namely, if is the symmetric matrix that defines the quadratic form, and is any invertible matrix such that is diagonal, then the number of negative elements in the diagonal of is always the same, for all such ; and the same goes for the number of positive elements.

This property is named after James Joseph Sylvester who published its proof in 1852.[1][2]

Statement

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Let   be a symmetric square matrix of order   with real entries. Any non-singular matrix   of the same size is said to transform   into another symmetric matrix  , also of order  , where   is the transpose of  . It is also said that matrices   and   are congruent. If   is the coefficient matrix of some quadratic form of  , then   is the matrix for the same form after the change of basis defined by  .

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

The number of  s, denoted  , is called the positive index of inertia of  , and the number of  s, denoted  , is called the negative index of inertia. The number of  s, denoted  , is the dimension of the null space of  , known as the nullity of  . These numbers satisfy an obvious relation

 

The difference,  , is usually called the signature of  . (However, some authors use that term for the triple   consisting of the nullity and the positive and negative indices of inertia of  ; for a non-degenerate form of a given dimension these are equivalent data, but in general the triple yields more data.)

If the matrix   has the property that every principal upper left   minor   is non-zero then the negative index of inertia is equal to the number of sign changes in the sequence

 

Statement in terms of eigenvalues

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The law can also be stated as follows: two symmetric square matrices of the same size have the same number of positive, negative and zero eigenvalues if and only if they are congruent[3] ( , for some non-singular  ).

The positive and negative indices of a symmetric matrix   are also the number of positive and negative eigenvalues of  . Any symmetric real matrix   has an eigendecomposition of the form   where   is a diagonal matrix containing the eigenvalues of  , and   is an orthonormal square matrix containing the eigenvectors. The matrix   can be written   where   is diagonal with entries  , and   is diagonal with  . The matrix   transforms   to  .

Law of inertia for quadratic forms

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In the context of quadratic forms, a real quadratic form   in   variables (or on an  -dimensional real vector space) can by a suitable change of basis (by non-singular linear transformation from   to  ) be brought to the diagonal form

 

with each  . Sylvester's law of inertia states that the number of coefficients of a given sign is an invariant of  , 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 positive definite (respectively, negative definite) have the same dimension. These dimensions are the positive and negative indices of inertia.

Generalizations

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Sylvester's law of inertia is also valid if   and   have complex entries. In this case, it is said that   and   are  -congruent if and only if there exists a non-singular complex matrix   such that  , where   denotes the conjugate transpose. In the complex scenario, a way to state Sylvester's law of inertia is that if   and   are Hermitian matrices, then   and   are  -congruent if and only if they have the same inertia, the definition of which is still valid as the eigenvalues of Hermitian matrices are always real numbers.

Ostrowski proved a quantitative generalization of Sylvester's law of inertia:[4][5] if   and   are  -congruent with  , then their eigenvalues   are related by   where   are such that  .

A theorem due to Ikramov generalizes the law of inertia to any normal matrices   and  :[6] If   and   are normal matrices, then   and   are congruent if and only if they have the same number of eigenvalues on each open ray from the origin in the complex plane.

See also

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References

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  1. ^ Sylvester, James Joseph (1852). "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" (PDF). Philosophical Magazine. 4th Series. 4 (23): 138–142. doi:10.1080/14786445208647087. Retrieved 2008-06-27.
  2. ^ Norman, C.W. (1986). Undergraduate algebra. Oxford University Press. pp. 360–361. ISBN 978-0-19-853248-4.
  3. ^ Carrell, James B. (2017). Groups, Matrices, and Vector Spaces: A Group Theoretic Approach to Linear Algebra. Springer. p. 313. ISBN 978-0-387-79428-0.
  4. ^ Ostrowski, Alexander M. (1959). "A quantitative formulation of Sylvester's law of inertia" (PDF). Proceedings of the National Academy of Sciences. A quantitative formulation of Sylvester's law of inertia (5): 740–744. Bibcode:1959PNAS...45..740O. doi:10.1073/pnas.45.5.740. PMC 222627. PMID 16590437.
  5. ^ Higham, Nicholas J.; Cheng, Sheung Hun (1998). "Modifying the inertia of matrices arising in optimization". Linear Algebra and Its Applications. 275–276: 261–279. doi:10.1016/S0024-3795(97)10015-5.
  6. ^ Ikramov, Kh. D. (2001). "On the inertia law for normal matrices". Doklady Mathematics. 64: 141–142.
  • Garling, D. J. H. (2011). Clifford algebras. An introduction. London Mathematical Society Student Texts. Vol. 78. Cambridge: Cambridge University Press. ISBN 978-1-107-09638-7. Zbl 1235.15025.
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  • Sylvester's law at PlanetMath.
  • Sylvester's law of inertia and *-congruence