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 eigenvaluesEdit
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 (, 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 formsEdit
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.
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: 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 : 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.
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^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. PMC222627. PMID 16590437.
^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.
^Ikramov, Kh. D. (2001). "On the inertia law for normal matrices". Doklady Mathematics. 64: 141–142.
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