Convexity of quadratic functions in two variables

We return to the example described here. We consider the two quadratic functions $p,q : mathbf{R}^2 rightarrow mathbf{R}$ , with values

$begin{array}{rcl} p (x) &=& 4x_1^2 + 2x_2^2 + 3 x_1x_2 + 4 x_1 + 5 x_2 + 2 times 10^5, q(x) &=& 4x_1^2 - 2x_2^2 + 3 x_1x_2 + 4 x_1 + 5 x_2 + 2 times 10^5. end{array}$

The Hessian of is independent of , and given by the constant matrix:

$nabla^2 p = left(begin{array}{cc} 8 & 3 3 & 4 end{array} right).$

We check that the eigenvalues of are positive, since the determinant, as well as the trace, of the above matrix are. Therefore, is convex.

Likewise, the Hessian of is

$nabla^2 q = left(begin{array}{cc} 8 & 3 3 & -4 end{array} right).$

This time, the Hessian has a negative eigenvalue, so is not convex.

Level sets and graph of the quadratic function . The epigraph is anything that extends above the graph in the -axis direction. This function is ‘‘bowl-shaped’’, or convex.

Level sets and graph of the quadratic function .