Negative DefiniteEdit

Negative definite

In linear algebra and its applications, a matrix is called negative definite when it defines a strictly negative quadratic form. Concretely, a real symmetric matrix A is negative definite if, for every nonzero vector x, the scalar x^T A x is negative. In the complex setting, the condition is x^* A x < 0 for all nonzero x, where A is Hermitian. This property is the counterpart to positive definiteness, which requires x^T A x > 0 for all nonzero x. The two notions play a central role in stability analysis, optimization, and the study of energy-like quantities in engineering and physics.

Negative definiteness is most often discussed for real symmetric matrices, because symmetry guarantees real eigenvalues and a well-behaved quadratic form. The defining inequality implies strong structural consequences for A and for the quadratic form it encodes. In particular, A must be invertible, since x^T A x cannot be negative for all x if zero were an eigenvalue.

In many texts, the property is stated equivalently via the spectrum: All eigenvalues of A are negative. Equivalently, −A is positive definite. This spectral perspective underpins computational methods and theoretical proofs alike, linking negative definiteness to diagonalization, norm bounds, and stability criteria. For those studying the geometric picture, the quadratic form x^T A x has a concave, downward-opening paraboloid shape in the real case, with level sets that shrink toward the origin.

Definitions and basic properties

Real and Hermitian setting: A is negative definite if x^T A x < 0 for all nonzero x; for complex spaces, A should be Hermitian so that x^* A x is real-valued. This guarantees a meaningful sign condition on the quadratic form.
Equivalences: A is negative definite if and only if all eigenvalues of A are negative; equivalently, −A is positive definite; equivalently, there exists a matrix P such that P^T A P = −I, reflecting a complete negative sign without loss of generality under congruence.
Invertibility and definiteness: Negative definiteness implies invertibility, since a zero determinant would force x^T A x = 0 for some nonzero x.
Comparison with related notions: Negative definite is the strict counterpart to positive definite; negative semidefinite and indefinite matrices relax or alter the sign behavior of the quadratic form. See also Positive definite and Indefinite matrix for related concepts.

Characterizations via Sylvester's criterion

For a real symmetric matrix A, there are several equivalent ways to certify negative definiteness. A classical route uses Sylvester's criterion adapted to the negative sign:

Leading principal minors alternate in sign: if δ_k denotes the determinant of the top-left k×k block of A (with δ_0 = 1 by convention), then A is negative definite if and only if δ_k has sign (−1)^k and δ_k > 0 for all k = 1, 2, ..., n.
If −A is positive definite, the standard positive definiteness criteria apply directly to −A, including the requirement that all leading principal minors of −A be positive.

These criteria are useful both in proofs and in numerical tests, and they illustrate how the sign of a matrix relates to the geometry of the associated quadratic form. See also Sylvester's criterion for a broader discussion of principal-minor tests.

Applications in optimization and analysis

Hessians and concavity: For a twice continuously differentiable function f, a negative definite Hessian at a point implies a strict local maximum there, since the second-order approximation is concave in every direction. In contrast, a positive definite Hessian indicates a local minimum. This connection between definiteness and curvature is a standard tool in optimization. See also Hessian matrix and Concave function.
Stability and Lyapunov methods: In dynamical systems, a symmetric matrix that is negative definite can serve as a Lyapunov function derivative under certain circumstances, providing certificates of stability. For linear systems x' = A x with A Hermitian, negative definiteness of A (i.e., all eigenvalues negative) ensures asymptotic stability. See also Lyapunov stability.
Energy and engineering: In structural engineering and physics, quadratic forms associated with negative definite matrices model energy-like quantities that decrease along trajectories, contributing to assessments of safety and reliability. The concept aligns with the broader idea that systems should dissipate or bound energy in stable configurations.
Computation and factorization: Negative definite matrices are closely related to positive definite matrices via A = −B with B positive definite. This duality allows one to transfer numerical methods, such as the Cholesky decomposition, from the positive-definite setting to the negative-definite setting by working with B = −A.

Examples and intuition

Simple example: A = [ [−2, 0], [0, −3] ] is negative definite, since for any nonzero x = [x1, x2]^T, x^T A x = −2 x1^2 − 3 x2^2 < 0.
Rank-one intuition: If A is negative definite, every nonzero direction in the space is assigned a strictly negative energy, reinforcing the notion of a stable, downward-curving quadratic form. This is in contrast to an indefinite matrix, where some directions yield positive energy and others yield negative energy.

Comparing negative definite with related notions

Positive definite: Reversing the sign changes the picture entirely; A is positive definite if x^T A x > 0 for all nonzero x. See Positive definite.
Negative semidefinite: If x^T A x ≤ 0 for all x and equality holds only at x = 0, A is negative semidefinite. The strictness of negative definiteness matters for guarantees of uniqueness and curvature.
Indefinite: If x^T A x takes both negative and positive values as x ranges over nonzero vectors, A is indefinite. This often signals directions of growth and decay in dynamical contexts.
Congruence invariance: The property of being negative definite is preserved under congruence transformations with invertible P; that is, P^T A P is negative definite if A is. This makes the sign of the quadratic form invariant under a change of basis, up to a scaling by the negative sign.