High Quality Content by WIKIPEDIA articles! 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 coordinates. Namely, if A is the symmetric matrix that defines the quadratic form, and S is any invertible matrix such that D = SAST 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.This property is named for J. J. Sylvester who published its proof in 1852.Let A be a symmetric square matrix of order n with real entries. Any non-singular matrix S of the same size is said to transform A into another symmetric matrix B = SAST, also of order n, where ST is the transpose of S. If A is the coefficient matrix of some quadratic form of Rn, then B is the matrix for the same form after the change of coordinates defined by S.

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