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How do you differentiate the following expressions with respect to the vector $x$.

I think I might be a little conceptually confused on what happens when you take the derivative with respect to a vector. What dimensions should you end with? For the problem, I am trying to solve, I think I should end up for each of these derivatives with the results in the derivative to be $R^{1\times1}$.

  1. $\frac{d}{dx}(b^TAx)$

  2. $\frac{d}{dx}(x^TAb)$

  3. $\frac{d}{dx}(x^TAx)$

where $b, x \in R^{n\times 1}$ and $A \in R^{n\times n}$

Also, if it helps A for this case is symmetric.


Update:

Thanks to the extra motivation by Mike and joriki, I think I now have them solved.

  1. $\frac{d}{dx}(b^TAx) = Ab$

  2. $\frac{d}{dx}(x^TAb) = Ab$

  3. $\frac{d}{dx}(x^TAx) = \textbf{A}\textbf{x} + \textbf{A}^T\textbf{x}$

But if anyone, would like to double check it that would be great.

  • 0
    Wikipedia's article on [Matrix calculus](http://en.wikipedia.org/wiki/Matrix_calculus) may help with the definition of vector derivatives.2011-10-19

2 Answers 2