Hessian of a quadratic function

Question

Assume that the quadratic function $f : {\Bbb R}^n \to {\Bbb R}$ is given by $$ f(x) = \frac12 x^T P x + q^T x + r$$ Calculate the Hessian of $f$.

Actually, I don't know how to apply Hessian on $f$ because Hessian requires 2nd order differentials and I don't know how to apply 2nd order differential on $f$.

$x^T=(x_1,\cdots,x_n)$. Expand $x^TPx$ as a sum of terms $p_{ij}x_i x_j$. — Martín-Blas Pérez Pinilla, Mar 26 '14 at 19:45

score 0 · Answer 1 · answered Mar 26 '14 at 20:40

0

Calculate partial derivative $\frac{\partial}{\partial x_i}\frac{\partial}{\partial x_j}$, put them in a matrix, this is the Hessian.

$$ \nabla^2 f(x) = \frac12(P+P^T) $$

answered Mar 26 '14 at 20:40

daw

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Hessian of a quadratic function

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