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.
, the number of positive, negative and zero eigenvalues (called the inertia of the matrix) of
This property is named after James Joseph Sylvester who published its proof in 1852.
be a symmetric square matrix of order
is the coefficient matrix of some quadratic form of
is the matrix for the same form after the change of basis defined by
can always be transformed in this way into a diagonal matrix
Sylvester's law of inertia states that the number of diagonal entries of each kind is an invariant of
, is called the positive index of inertia of
, is called the negative index of inertia.
These numbers satisfy an obvious relation The difference,
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.)
has the property that every principal upper left
is non-zero then the negative index of inertia is equal to the number of sign changes in the sequence 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] (
The positive and negative indices of a symmetric matrix
are also the number of positive and negative eigenvalues of
is an orthonormal square matrix containing the eigenvectors.
-dimensional real vector space) can by a suitable change of basis (by non-singular linear transformation from
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
-congruent if and only if there exists a non-singular complex matrix
In the complex scenario, a way to state Sylvester's law of inertia is that if
-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
A theorem due to Ikramov generalizes the law of inertia to any normal matrices
are congruent if and only if they have the same number of eigenvalues on each open ray from the origin in the complex plane.