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
StatementFor a symmetric matrix S and any invertible matrix P, the numbers of positive, negative, and zero eigenvalues of PSP^T are constant. 1 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
In Branch
Source Sylvester's law of inertia, Wikipedia
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.
In Branch: Algebra, Lead sentence
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic f
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