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 $f^\prime(x) = Qx - b$?

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Can you explain your notation a little better? – Ian Coley Mar 29 '13 at 17:46
I'm really not recognizing your definition of a quadratic form at all. It would help to include that. Definitely say what $Q$ and $b$ are (constants, apparently?) – rschwieb Mar 29 '13 at 17:47
Sorry, Q is a symmetric matrix and x and b are n-dimensional vectors. It should be x^tQx, for example. What I meant by the last statement was that it seems this 1/2 comes from the 1-D case. Here Q is a scalar (from a matrix) and b a scalar (from a vector) – user67218 Mar 29 '13 at 18:00

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 $$f(x,y,z) = x^2 + y^2 + z^2 + y z + z x + x y.$$
@StephenBosch, just try a two dimensional example, $f(x,y)= x^2 + xy + y^2.$ There is no need for the 1/2, it is a choice. – Will Jagy Feb 6 '14 at 20:42