A real symmetric matrix can be diagonalised by an orthogonal transformation. Using coordinates along the principal axes (eigenvectors), in dimensions

so the sign of the quadratic form is determined entirely by the eigenvalues:

• is positive definite iff all eigenvalues satisfy ;

• is negative definite iff all eigenvalues satisfy ;

• if all eigenvalues are non-zero but of mixed signs, the quadratic form takes both signs, and is indefinite;

• if any eigenvalue is zero, the quadratic form does not determine definiteness, and higher order terms in the Taylor series are needed to classify the stationary point.

Suppose the Hessian matrix of at a stationary point has non-zero eigenvalues, so that in coordinates aligned with its principal axes

On contours near , where is constant,

• At a maximum or minimum, and have the same sign, and the contours are ellipses.

• At a saddle point, and have opposite signs, and the contours are hyperbolae.

In particular, a Hessian that is positive definite at makes a local minimum, a negative definite one a local maximum, while an indefinite Hessian may give a maximum, minimum or saddle point.