Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
1  
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

1 Answer 1

up vote 3 down vote accepted

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 $$ f(x,y,z) = x^2 + y^2 + z^2 + y z + z x + x y. $$

share|improve this answer
    
Thank you, that makes sense! –  user67218 Mar 30 '13 at 18:34
    
Forgive me, I have the same question but I'm struggling to grasp your explanation. I still don't see why there needs to be a 1/2 in front of the first term. The quadratic form as defined in all the textbooks I've seen is simply x^tQx. Is there a place where I can find this derived in detail? How am I to recognize that Q is a Hessian matrix of second partials? –  Stephen Bosch Feb 6 at 20:38
1  
@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 at 20:42

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.