For quadratic functions, the Hessian (matrix of second-derivatives) is a constant matrix, that is, it does not depend on the variable [latex]x[/latex].

As a specific example, consider the quadratic function

[latex]\begin{align*} q(x) &= 8x_1^2 + 6x_1x_2 + 4x_2^2 -6x_1 +9x_2 + 10. \end{align*}[/latex]

The Hessian is given by

[latex]\begin{align*} \frac{\partial^2 q}{\partial x_i \partial x_j}(x) &= \left(\begin{array}{cc} \dfrac{\partial^2 q}{\partial x_1^2}(x) & \dfrac{\partial^2 q}{\partial x_1 \partial x_2}(x) \\[3ex] \dfrac{\partial^2 q}{\partial x_2 \partial x_1}(x) & \dfrac{\partial^2 q}{\partial x_2^2}(x) \end{array}\right) = 2\left(\begin{array}{ll} 8 & 3 \\ 3 & 4 \end{array}\right) \text{. } \end{align*}[/latex]