Sylvester's law of inertia

For a real quadratic form, the counts of positive, negative, and zero terms in a diagonal representation are invariant under invertible linear changes of variables.

Quadratic form: A homogeneous polynomial of degree two in several variables, commonly represented by a symmetric matrix. The law classifies the signs of a quadratic form after diagonalization.
Spectral theorem: The theorem that a real symmetric matrix has an orthonormal basis of eigenvectors. It diagonalizes the matrix of a real quadratic form using an orthogonal change of variables.
James Joseph Sylvester: An English mathematician who made foundational contributions to matrix theory, invariant theory, and number theory. The law bears his name and appeared in his work on quadratic forms.
Hessian matrix: The matrix of second partial derivatives of a scalar-valued function. At a critical point, its inertia determines the local quadratic behavior of the function.
Sylvester's law of inertia for Hermitian forms: The extension of inertia invariance to complex Hermitian forms under invertible congruence. It carries the same invariant principle from real quadratic forms to complex Hermitian forms.
Symmetric matrix: A square matrix equal to its transpose. Every real quadratic form can be represented by a symmetric matrix.
Eigenvalue: A scalar λ for which a linear transformation sends some nonzero vector v to λv. The signs of the eigenvalues give the positive and negative counts in an orthogonal diagonalization.
Quadratic form (history): The historical development of mathematical expressions of degree two and their classification. Classifying quadratic forms supplied the problem that the law resolves.
Morse theory: A branch of topology relating a manifold's shape to the critical points of smooth functions. The Hessian's negative count gives the Morse index used to track topology.
Similarity transformation: A matrix transformation sending A to P⁻¹AP for an invertible matrix P. Similarity preserves eigenvalues, whereas congruence preserves quadratic-form inertia.

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Bilinear formRelated: Its signature classifies real symmetric forms up to change of basis.