Hessian of a quadratic function

L. El Ghaoui

Hessian of a quadratic function

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

As a specific example, consider the quadratic function

$q(x) = 8x_1^2 6x_1x_2 + 4x_2^2 -6x_1 +9x_2 + 10.$

The Hessian is given by

$\frac{\partial^2 q}{\partial x_i \partial x_j}(x)=\left(\begin{array}{cc} \frac{\partial^2 q}{\partial x_1^2}(x) & \frac{\partial^2 q}{\partial x_1 \partial x_2}(x) \\ \frac{\partial^2 q}{\partial x_2 \partial x_1}(x) & \frac{\partial^2 q}{\partial x_2^2}(x) \end{array}\right)=2\left(\begin{array}{ll} 8 & 3 \\ 3 & 4 \end{array}\right) \text {. }$

Hessian of a quadratic function

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