Linear Algebra

Quadratic Forms and Definiteness

Read a symmetric matrix as a function, and the spectral theorem sorts it by sign.

Updated July 2026

A symmetric matrix can be read two ways. The spectral theorem notes treated it as an operator, something that acts on vectors and has a clean eigenbasis. There is a second reading, just as useful, in which the same matrix is a function that eats a vector and returns a number:

Q (x) = x^{⊤} A x = i, j \sum A_{ij} x_{i} x_{j} .

This is a quadratic form, the natural generalisation of $a x^{2}$ to many variables. It appears wherever an energy, a variance, or a distance shows up, and the question you almost always want to ask of it is: what is its sign?

Diagonalising the form

Because $A$ is symmetric, the spectral theorem gives an orthonormal basis of eigenvectors and real eigenvalues, $A = Q Λ Q^{⊤}$ . Change to that basis, $y = Q^{⊤} x$ , and the cross terms vanish:

Q (x) = i \sum λ_{i} y_{i}^{2} .

In its own principal axes the form is just a weighted sum of squares, weighted by the eigenvalues. Its sign is therefore decided, completely, by the signs of the $λ_{i}$ .

Definition

Definiteness

$A$ is positive definite if $Q (x) > 0$ for every non-zero $x$ , which holds exactly when all eigenvalues are positive. It is negative definite when all are negative, and indefinite when both signs appear. The borderline cases, with a zero eigenvalue, are called semidefinite.

A criterion you can check by hand

Computing every eigenvalue is more than you need just to know a sign. Sylvester's criterion reads definiteness off the leading principal minors, the determinants of the top-left $k \times k$ blocks:

A is positive definite ⟺ det A_{k} > 0 for all k .

For a $2 \times 2$ symmetric matrix this is just: the top-left entry positive and the whole determinant positive. No eigenvalues required.

Where the sign is the whole story

The reason definiteness earns its own name is that it answers a geometric question about a critical point. The second-order behaviour of any smooth function near a stationary point is a quadratic form, its Hessian, and the sign of that form is what a minimum, a maximum and a saddle are.

Consequence

The second-derivative test, honestly

At a stationary point the Hessian's definiteness classifies it: positive definite is a minimum, negative definite a maximum, indefinite a saddle. The one-variable rule about the second derivative being positive is this statement with a single eigenvalue. When the Hessian is only semidefinite the test is silent, which is not a gap in the theory but the truthful statement that the quadratic term alone cannot decide.

The same idea runs through mechanics as stability of an equilibrium, through statistics as the positive-definiteness of a covariance matrix, and through the last thermodynamics note as the convexity that forces the response functions positive. In each case a symmetric matrix is standing in for a bowl, and definiteness is asking which way the bowl opens.