A quadratic form associates a scalar with a vector through a matrix:

$$q(x)=x^TAx.$$

For a symmetric matrix $A$, the sign of this quantity in different directions describes the matrix's definiteness.

$A$ is positive definite when

$$x^TAx>0$$

for every nonzero $x$, and positive semidefinite when

$$x^TAx\ge0$$

for every $x$.

Negative definite and negative semidefinite matrices reverse these inequalities. A symmetric matrix is indefinite when its quadratic form is positive in some directions and negative in others.

Quadratic forms arise naturally in energy, least squares and second-order approximations. For a Hessian, definiteness describes whether the local curvature bends upward, downward or in different directions at once.