Gradient function using matrix notation

[hotvette]

I think I'm having a brain freeze. I'm trying to determine grad f where f(x) = 1/2 x^TQx + q^Tx. I can get to the point where df = (x^TQ + q^T)dx, but I don't know how to get to the final result grad f = Qx + q.

Can someone explain it?

 

[tiny-tim]

hi hotvette!

Hint: use x^TQx = ∑_ij Q_ijx_ix_j, q^Tx = ∑_i q_ix_i

 

[quasar987]

What is df = (xTQ + qT)dx supposed to mean?

Using the definition of the derivative Df, I calculated that the matrix of Df (that is, grad(f)) is, is ½(Qx+xQ)+q. If Q is a symmetric matrix, then xQ=Qx and we find your result grad(f)=Qx+q.

 

[hotvette]

tiny-tim and quasar987,

Thanks for your replies. The calculus classes I had (l-o-n-g time a go) never used matrix notation, so I'm trying to learn it on my own. I've been using this:

http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/calculus.html#deriv_linear

as a guide and was following the terminology in the link. I'm treating dx as a vector quantity (e.g. dx = [dx₁, dx₂, ..., dx_n]^T). I'm just having a hard time deriving the expression for grad f based on the info in the link. Also, sorry, but I don't see where the sum notation leads.

 

[tiny-tim]

hi hotvette!

I've been using this:

http://www.ee.ic.ac.uk/hp/staff/dmb/matrix/calculus.html#deriv_linear

as a guide and was following the terminology in the link.

eugh! :yuck: I've never come across that terminology before, and I find it really confusing.

Personally, I'd use some other book.

What do you think, quasar987? ​

 

[hotvette]

eugh! :yuck: I've never come across that terminology before, and I find it really confusing

I don't follow it very well either. I'd be delighted to use an understandable alternative. Are there any online references you'd recommend?