Hessian of a quadratic function

Symmetric Matrices > Definitions > Example

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

As a specific example, consider the quadratic function

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

The Hessian is given by

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