TITLE: Derivative of quadratic matrix form with respect to the matrix
QUESTION [11 upvotes]: Suppose we have the following quadratic form
$$
f(M)=x^TMx
$$
where $f: \mathbb{R}^{n \times m} \rightarrow \mathbb{R}$, and $x \in \mathbb{R}^n$.
As it is obvious, this function is linear in $M$. What is the derivative of $f(M)$ with respect to $M$. I think it is $xx^T$ but why?

REPLY [5 votes]: The strategy is to write the expression as a scalar using index notation, take the
derivative, and re-write in matrix form. 
Note that to write the function as a summation of matrices we have to write just one scalar as a matrix multiplication because the function is scalar:
$$
f(M)= [x^TMx]_{11}= \sum_i x_{i1}[Mx]_{i1}
$$
At first summation, we have $x_{i1}$ because we have $x^T$ as the first term.
Now expand $[Mx]_{i1}$
$$
f(M)= [x^TMx]_{11}= \sum_i x_{i1}\sum_j M_{ij}x_{j1}=\sum_i \sum_j x_{i1} M_{ij}x_{j1}
$$
Now take the derivative with respect to $M_{ij}$
$$
\frac{\partial f(M)}{\partial M_{ij}}=\sum_i \sum_j x_{i1}x_{j1}
$$
Looking at the indices, we can see that
$$
\sum_i \sum_j x_{i1}x_{j1}=\sum_i \sum_j x_{j1}x_{i1}=[xx^T]_{ji}
$$
Therefore,
$$
\frac{\partial f(M)}{\partial M_{ij}}=xx^T
$$