# Proof of derivative of $x^TBx$ using the product rule

I'm trying to prove that when $$f(x) =x^TBx$$, then $$f'(x) = (B + B^T)x$$. I haven't found this formula online but going through the calculations using index notation this is what I came up with. This would simplify to $$2Bx$$ when $$B$$ is symmetric. The accepted response to this discussion says that the solution is actually $$f'(x) = x^T(B + B^T)$$, going through the proof there, I see how he got there but I can't see where the mistake is in mine then.

The setup

• $$x \in \mathbb{R^n}$$, it is always a column vector
• $$B \in \mathbb{R^{n \times n}}$$, $$B$$ may not be symmetric

My approach

Let $$g(x)=x^TB$$ and $$h(x)=x$$, then I can write $$f(x)=g(x)h(x)$$. Then

• $$f(x) \in \mathbb{R}$$
• $$g(x) \in \mathbb{R^{1 \times n}}$$
• $$h(x) \in \mathbb{R^n}$$
• $$f'(x) \in \mathbb{R^n}$$
• $$g'(x) \in \mathbb{R^{n \times n}}$$
• $$h'(x) \in \mathbb{R^{n \times n}}$$

I've gone through myself why $$g'(x) = B$$ and $$h'(x) = I_n$$, so I won't go through those here.

Then, using the product rule I get:

$$f'(x) = g'(x)h(x) + g(x)h'(x)$$

The problem is that the dimensions don't add up. I get $$g'(x)h(x) = Bx \in \mathbb{R^{n}}$$, which is good. However, I also have $$g(x)h'(x) = x^TBI_n = x^TB \in \mathbb{R^{1 \times n}}$$ and as far as I know I can't add up two vectors of different sizes.

I know that the solution is going to be the transpose of second term, I just can't seem to find where that transpose would come from.

Why do I need to take the transpose of the second term?

: Please don't reply with a different proof. What I'm looking for is to understand where I made the mistake in my calculation because obviously I made a step which was incorrect and without understanding where that is I'm likely to make that mistake again.

• Take a look at this. – Rodrigo de Azevedo Apr 20 at 22:46
• @RodrigodeAzvedo That seems to be a completely different proof from what I have gone through. – norbertk Apr 21 at 15:13
• It's very simple, though. – Rodrigo de Azevedo Apr 21 at 15:15
• It does seem so but not sure how that points to the error I made in mine. – norbertk Apr 21 at 15:32

It may be useful to introduce the Frobenius inner product as:

$$A:B = \operatorname{tr}(A^TB)$$

with the following properties derivied from the underlying trace function

\eqalign{A:BC &= B^TA:C\cr &= AC^T:B\cr &= A^T:(BC)^T\cr &= BC:A \cr }

Then we work with differentials to find the gradient. The product rule works as you expect. At each side of the colons, you can note that dimensions are consistent.

\eqalign{ f&= x^TBx\\ &= x : Bx\\ df &= dx : Bx + x : Bdx\\ &=dx : Bx + B^Tx:dx\\ &=(B + B^T)x : dx }

Thus the gradient is: $$\frac{\partial f}{\partial x} =(B + B^T)x$$

edit:

The issue is that for vector terms: $$\frac{\partial(u^Tv)}{\partial x} \ne \Bigg(\frac{\partial u^T}{\partial x}\Bigg)v + u^T\Bigg(\frac{\partial v}{\partial x}\Bigg)$$

When working with differentials, on the other hand, it holds:

$$d(A\star B) = dA\star B + A\star dB$$

where $$\star$$ can be Frobenius, Kronecker, Hadamard, matrix product, etc.

If you work the differential with the matrix product form you will see that a term $$dx^T$$ appears. Dealing with this term for grouping dx is what causes your missing transpose to appear.

If you want to directly apply a product rule it should read:

$$\frac{\partial(u \cdot v)}{\partial x} = \Bigg(\frac{\partial u}{\partial x}\Bigg)^T v + \Bigg(\frac{\partial v}{\partial x}\Bigg)^T u$$

where $$u \cdot v = u^Tv$$, with $$u=x$$ and $$v = Bx$$.

• I will definitely look up more about Frobenius inner product and also about differentials, but I don't see how this answer points to where I made the mistake in my calculation. – norbertk Apr 21 at 15:07
• I've added some clarification in the edit. – Traws Apr 22 at 3:23
• I still don't understand where that transpose come from. I accepted the answer because it does contain several things that are related to the question but I will look for other resources that explain clearly the dimensions of objects that result from differentiating matrices. – norbertk Apr 25 at 10:49

The easiest way is really to right the function as a scalar :

$$f(x)=x^TBx=\sum_{i=1}^n\sum_{j=1}^n x_ib_{ij}x_j.$$ Then $$\partial _kf(x)=\sum_{i=1}^n x_ib_{ik}+\sum_{i=1}^n b_{ki}x_i=\sum_{i=1}^n(b_{ik}+b_{ki})x_i.$$ Therefore, $$f'(x)=x^T(B+B^T)=(B^T+B)x.$$

• Thanks, I have gone through the same derivation myself and got to the result that I posted. The question I asked was about the use of product rule and the mistake in my calculation. The equality in the "therefore" part isn't clear though just from the derivation, but J.G.'s answer explains that it is just a convention, so I can just accept that fact. – norbertk Apr 20 at 12:06

The way you try to define derivatives with respect to $$x$$ has a subtle inconsistency. On the one hand you insist the derivative of $$x^TB$$ is $$B$$, implying differentiation's effect is to cancel an $$X^T$$ from the left. On the other hand, you insist the derivative of $$X$$ (i.e. $$IX$$, not $$X^TI=X^T$$) is $$I$$, i.e. differentiation cancels an $$X$$ from the right. It's better to work through explicit indices.

Whether the derivative of the scalar $$x^TBx$$ is the column vector $$(B+B^T)x$$ or the transpose thereof, the row vector $$x^T(B+B^T)$$, is a matter of convention. (However, a column-vector convention has one obvious advantage: the chain rule becomes $$df=(dx)^T\nabla f$$.) What is not a matter of convention is its $$i$$th component is $$\partial_i (x_jB_{jk}x_k)=\delta_{ij}B_{jk}x_k+x_jB_{jk}\delta_{ik}=B_{ik}x_k+B_{ji}x_j=[(B+B^T)x]_i.$$

• Thanks for the reply. I understand your derivation, but I don't know how to retrace to the product rule in my original example. I get that you are using the product rule but it still seems to me as adding up vectors with different sizes. If I use the same convention (row vs column vector results) when calculating every derivative in my result, I wouldn't expect having to transpose the terms just so that the sizes match up. – norbertk Apr 20 at 12:09
• @norbertk I've edited my answer to address your own calculation. – J.G. Apr 20 at 12:21
• I understand it's simpler to work with indices and it seems that I did arrive to the right solution myself using indices. However, the whole point of me asking this question was about understanding where I made the mistake in the original calculation and not about coming with different methods to calculate the same thing. I want to understand what step is incorrect. You noted that there is an inconsistency in how I define derivatives. This seems to be the problem I'm looking for. I feel that the issue may be with $x^TB$, because when I differentiate that I could get an n x 1 x n tensor? – norbertk Apr 21 at 15:05