Indefinite quadratic form ($q(u)>0,q(v)<0$)

A real quadratic form is indefinite when it takes both positive and negative values. For a real symmetric matrix representing it, this means that its eigenvalues include both signs. For example, $q(x,y)=x^2-y^2$ has $q(1,0)=1$ and $q(0,1)=-1$; a zero value at the nonzero vector $(1,1)$ does not imply positive semidefiniteness.^[ref]

A real symmetric matrix $A$ is indefinite when $x^{T}Ax$ is an indefinite quadratic form. Equivalently, $A$ has both a positive and a negative eigenvalue, by the spectral theorem. A zero eigenvalue is permitted, so an indefinite matrix need not be an invertible matrix.