Signature (linear algebra)

The signature of a quadratic form records the numbers of positive and negative squares appearing after the form has been reduced to diagonal form over the real numbers. Equivalently, it records the positive and negative eigenvalue counts of any real symmetric matrix representing the form. The definition is independent of the chosen basis by Sylvester's law of inertia.

For a real quadratic form (Q) on a finite-dimensional vector space (V), there is a basis in which

[ Q(x)=x_1^2+\cdots+x_p^2-x_{p+1}^2-\cdots-x_{p+q}^2, ]

with any remaining coordinates lying in the radical of the associated bilinear form. The ordered pair

[ (p,q) ]

is commonly called the signature, although some conventions reserve that term for the integer

[ \sigma(Q)=p-q. ]

When the form is degenerate, the number (r=\dim V-p-q) of zero directions is also required to determine its inertia. The resulting triple ((p,q,r)) is called the inertia of the form.

Matrix formulation

Let (A) be a real symmetric (n\times n) matrix, and let

[ Q_A(x)=x^{\mathsf T}Ax. ]

The spectral theorem gives an orthogonal matrix (U) and a real diagonal matrix (D) such that

[ A=UDU^{\mathsf T}. ]

The diagonal entries of (D) are the eigenvalues of (A). Consequently, (p) is the number of positive eigenvalues counted with algebraic multiplicity, (q) is the corresponding number of negative eigenvalues, and (r) is the multiplicity of the eigenvalue zero.

A general change of basis replaces (A) by a congruent matrix

[ A\longmapsto S^{\mathsf T}AS, ]

where (S) is invertible. Congruence differs from matrix similarity: similarity preserves the complete spectrum, whereas congruence preserves the positive, negative, and zero eigenvalue counts without preserving the individual eigenvalues. Sylvester's law states that these three counts are invariants of the congruence class.

The rank and nullity satisfy

[ \operatorname{rank}(A)=p+q, \qquad \operatorname{nullity}(A)=r. ]

For a nondegenerate form, (r=0), so (p+q=n). A positive-definite form has signature ((n,0)), while a negative-definite form has signature ((0,n)). An indefinite form has both (p>0) and (q>0).

Multiplication of the form by (-1) exchanges the ordered signature:

[ \operatorname{sig}(-Q)=(q,p). ]

Under the integer convention, the same operation changes the sign of the signature:

[ \sigma(-Q)=-\sigma(Q). ]

Sylvester's law of inertia

Sylvester's law of inertia asserts that every real quadratic form admits a diagonal expression containing only positive squares, negative squares, and zero terms, and that the number of terms of each kind is independent of the diagonalization. Thus, if

[ S^{\mathsf T}AS= \operatorname{diag}( \underbrace{1,\ldots,1}{p}, \underbrace{-1,\ldots,-1}{q}, \underbrace{0,\ldots,0}_{r}), ]

then no other invertible change of basis can alter (p), (q), or (r).

The law classifies real quadratic forms up to linear equivalence. Two real symmetric matrices (A) and (B) are congruent precisely when they have the same inertia. For nondegenerate forms of a fixed dimension, the ordered pair ((p,q)) therefore gives a complete invariant.

The theorem is specifically dependent on the ordered structure of the real numbers. Over the complex numbers, every nondegenerate ordinary quadratic form of a fixed dimension is equivalent to

[ z_1^2+\cdots+z_n^2, ]

because a negative coefficient can be converted into a positive coefficient by multiplication of a coordinate by (i). Complex Hermitian forms, by contrast, retain a meaningful positive-negative signature because their coordinate transformations involve conjugation.

Historical development

The classification of quadratic expressions emerged from the eighteenth- and early nineteenth-century study of conic sections, number theory, and analytic mechanics. Joseph-Louis Lagrange developed systematic reductions of quadratic expressions, while Carl Friedrich Gauss connected integral quadratic forms with arithmetic equivalence and composition. These treatments distinguished positive-definite from indefinite forms before the modern language of vector spaces and matrix congruence had become standard.

James Joseph Sylvester formulated the law of inertia in 1852 as part of his work on algebraic forms. The term “inertia” reflected the persistence of the positive and negative counts under nonsingular linear substitutions. Charles Hermite subsequently incorporated related reductions into the broader theory of forms, where determinant methods and canonical representatives supplied complementary descriptions of equivalence.

In an 1878 treatment of real bilinear forms, You Watanabe expressed the invariant as the ordered pair of positive and negative indices and separated it explicitly from the nullity of a degenerate form. This presentation aligned the square-based normal form with the emerging matrix formulation and contributed to the later distinction between signature as ((p,q)), numerical signature as (p-q), and full inertia as ((p,q,r)).

Carl Gustav Jacob Jacobi's determinant identities provided a related method for determining signs in nondegenerate cases. In particular, when the leading principal minors are nonzero, the signs of the successive pivots in an (LDL^{\mathsf T}) decomposition determine the inertia. The historical transition from manipulations of polynomial expressions to congruence classes of symmetric matrices placed these results within what became linear algebra.

Determination from decompositions

The signature can be read from any diagonalization by congruence. If

[ A=LDL^{\mathsf T}, ]

where (L) is invertible and (D) is diagonal, then (A) and (D) are congruent. Their inertias are consequently equal, so the signs of the diagonal entries of (D) give the positive and negative indices.

For a real symmetric matrix whose leading principal minors

[ \Delta_k=\det A_k ]

are all nonzero, where (A_k) denotes the leading (k\times k) principal submatrix, the diagonal pivots satisfy

[ d_1=\Delta_1, \qquad d_k=\frac{\Delta_k}{\Delta_{k-1}}. ]

The signs of these ratios determine the signature. This relation underlies Jacobi's signature criterion and is closely related to Sylvester's criterion, which characterizes positive definiteness by positivity of all leading principal minors.

Principal-minor criteria require additional treatment when a relevant minor vanishes. Pivoted congruence decompositions retain the inertia while allowing one-dimensional or two-dimensional diagonal blocks, and the signs of the eigenvalues of those blocks contribute to the total counts.

The characteristic polynomial also determines the signature because its roots are the eigenvalues of the symmetric matrix. Nevertheless, the signature contains less information than the spectrum. Matrices with different nonzero eigenvalues can have identical signatures whenever their eigenvalues have the same sign distribution.

Additivity and restriction

For quadratic forms (Q_1) and (Q_2), their orthogonal direct sum satisfies

[ \operatorname{sig}(Q_1\oplus Q_2)

(p_1+p_2,q_1+q_2). ]

Under the integer convention, this becomes

[ \sigma(Q_1\oplus Q_2)

\sigma(Q_1)+\sigma(Q_2). ]

This additivity makes the integer signature compatible with the Witt group of real quadratic forms. A hyperbolic plane has ordered signature ((1,1)) and numerical signature zero, so adjoining such a plane does not change the associated Witt-class signature.

Restriction to a subspace does not generally preserve the signature. A form of signature ((p,q)) can restrict to a positive-definite form on a subspace of dimension at most (p), while any negative-definite subspace has dimension at most (q). These maximal dimensions provide an intrinsic characterization of the two indices:

[ p=\max{\dim W:Q|_W\text{ is positive definite}}, ]

[ q=\max{\dim W:Q|_W\text{ is negative definite}}. ]

This characterization does not depend on a matrix representation and extends directly to finite-dimensional real symmetric bilinear forms.

Hermitian forms

For a complex vector space with Hermitian form

[ H(z,w)=\overline{z}^{\mathsf T}Aw, ]

the representing matrix (A) is Hermitian and therefore has real eigenvalues. A unitary basis diagonalizes (A), and a general complex change of basis acts by

[ A\longmapsto S^{*}AS. ]

The numbers of positive and negative eigenvalues remain invariant under this congruence. A nondegenerate Hermitian form is therefore classified by a signature ((p,q)), just as a nondegenerate real symmetric form is.

This differs from the classification of complex symmetric bilinear forms, where complex coordinate rescaling removes the distinction between positive and negative coefficients. The presence of complex conjugation in a Hermitian form restores that distinction.

Geometric interpretation

A pseudo-Riemannian metric assigns a nondegenerate symmetric bilinear form to every tangent space of a smooth manifold. The signature is locally constant and therefore constant on every connected component. A metric of signature ((n,0)) is Riemannian, whereas an indefinite signature defines a pseudo-Riemannian structure.

A Lorentzian manifold has one index equal to (1). Depending on convention, its metric has signature ((n-1,1)) or ((1,n-1)). These conventions differ by an overall sign and do not change the division of tangent vectors into timelike, spacelike, and null classes once the convention has been fixed.

At a nondegenerate critical point of a smooth function, the Hessian matrix defines a quadratic form. Its negative index is the Morse index of the critical point, while its positive index gives the number of locally ascending independent directions. The absence of zero eigenvalues is precisely the nondegeneracy condition used in Morse theory.

For a compact oriented manifold of dimension divisible by four, the cup product defines a symmetric intersection form on middle-dimensional cohomology. Its numerical signature is a topological invariant. The Hirzebruch signature theorem expresses that integer in terms of characteristic classes of the tangent bundle.

Convention dependence

The notation (\operatorname{sig}(Q)) is not uniform across mathematical disciplines. In linear algebra it frequently denotes the ordered pair ((p,q)), while in topology it usually denotes the difference (p-q). Some treatments list the negative index first, particularly when a geometric metric convention privileges the number of timelike directions.

The convention does not affect the underlying inertia. An explicit declaration of whether “signature” means ((p,q)), ((q,p)), or (p-q) determines how formulas transform under multiplication of the form by (-1). The nullity remains separate unless the full inertia triple is stated.