Quadratic Forms

Question

When defining a quadratic form why is it that we place $\frac{1}{2}$ in front? That is, why do we use $f(x) = \frac{1}{2}(x^T Qx) - b^T x$? Is this simply a convention that comes from the one-dimensional case where we would have $f^\prime(x) = Qx - b$?

Answer 1

This way $Q$ is the Hessian matrix of second partials. Also, it allows $Q$ to have all integer entries in some special cases of interest, such as $$ f(x,y,z) = x^2 + y^2 + z^2 + y z + z x + x y. $$