Theorem

Sylvester's Law of Inertia

Algebra

Sylvester's Law of Inertia states that although a real quadratic form can be diagonalized in many different ways by a change of variables, the number of positive, negative and zero coefficients that appear on the diagonal is always the same regardless of which diagonalization is chosen. Named for James Joseph Sylvester, this invariant triple is called the form's signature, and the theorem underlies the classification of quadratic forms and symmetric bilinear forms over the real numbers.

Facts

Statement

For a symmetric matrix S and any invertible matrix P, the numbers of positive, negative, and zero eigenvalues of PSP^T are constant. 1

Proof Year

1852 1

Classification

Statement Form

Classification Theorem 1

Sources

1. Sylvester's law of inertia, Wikipedia

Lead, first paragraph
Namely, if S is a symmetric matrix, then for any invertible matrix P, the numbers of positive, negative, and zero eigenvalues of S' = PSP^T are constant (i.e., the inertia of S' is constant).
Lead, naming sentence
This property is named after James Joseph Sylvester, who published its proof in 1852.

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Sylvester's Law of Inertia