Exam

Find necessary and sufficient conditions on the quadratic form

a x^{2} + 2 b x y + c y^{2}

R^{2}

such that the corresponding symmetric bi-linear form defines a Lorentz product on

R^{2}

The matrix of the Gramian of the canonical basis

e_{1}, e_{2}

R^{2}

is given by

(\begin{array}{cc} a & b \\ b & c \end{array})

and the determinant is strictly negative iff

a c - b^{2} < 0