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 S {\displaystyle S} is a symmetric matrix, then for any invertible matrix P {\displaystyle P} , the numbers of positive, negative, and zero eigenvalues of S ′ = P S P T {\displaystyle S'=PSP^{\mathsf {T}}} are constant (i.e., the inertia of S ′ {\displaystyle S'} is constant). This result is particularly useful when S ′ {\displaystyle S'} is diagonal, as the inertia of a diagonal matrix can easily be obtained by looking at the signs of its diagonal elements. This property is named after James Joseph Sylvester, who published its proof in 1852.
Statement Let S {\displaystyle S} be a symmetric square matrix of order n {\displaystyle n} with real entries. Any non-singular square matrix P {\displaystyle P} of the same order is said to transform S {\displaystyle S} into another symmetric matrix S ′ = P S P T {\displaystyle S'=PSP^{\mathsf {T}}} , also of order n {\displaystyle n} , where P T {\displaystyle P^{\mathsf {T}}} is the transpose of P {\displaystyle P} . It is also said that matrices S {\displaystyle S} and S ′ {\displaystyle S'} are congruent. If S {\displaystyle S} is the coefficient matrix of some quadratic form on R n {\displaystyle \mathbb {R} ^{n}} , then S ′ {\displaystyle S'} is the coefficient matrix of the same form after the change of basis defined by P {\displaystyle P} . A symmetric matrix S {\displaystyle S} can always be transformed in this way into a diagonal matrix D {\displaystyle D} which has entries only 0 {\displaystyle 0} , + 1 {\displaystyle +1} , − 1 {\displaystyle -1} along the diagonal. Sylvester's law of inertia states that the number of diagonal entries of each kind is an invariant of S {\displaystyle S} , i.e., does not depend on the matrix P {\displaystyle P} used. The number of + 1 {\displaystyle +1} 's, denoted n + {\displaystyle n_{+}} , is called the positive index of inertia of S {\displaystyle S} , and the number of − 1 {\displaystyle -1} 's, denoted n − {\displaystyle n_{-}} , is called the negative index of inertia of S {\displaystyle S} . The number of 0 {\displaystyle 0} 's, denoted n 0 {\displaystyle n_{0}} , is the dimension of the null space of S {\displaystyle S} , known as the nullity of S {\displaystyle S} . These numbers satisfy the obvious relation
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