Q:

Derivative of a trace norm or a Frobenius norm of a matrix wrt to a vector

I have a Frobenius norm of a multiplication of two matrices
$||U^TL^\frac{1}{2}||_F^2$
the $L$ matrix is a symmetric matrix, so it can be written as 
$tr(U^TLU)$
Now I want to get the derivative of this norm wrt a row vector in $U$, which is $U_i.$ as a vector. Note $U$ is $m \times n$, and $L$ is $m\times m$.

A:

Hint: With block matrix multiplication, we can rewrite
$$
U^TLU = \pmatrix{U_1^T & \cdots & U_m^T}\pmatrix{L_{11} & \cdots & L_{m1}\\ \vdots & \ddots & \vdots\\ L_{m1} & \cdots & L_{mm}} \pmatrix{U_1\\ \vdots \\U_m} = 
\sum_{i,j = 1}^m L_{i,j} U_i^T U_j
$$
So, we have
$$
\operatorname{Tr}(U^TLU) = \operatorname{Tr}\left[\sum_{i,j = 1}^m L_{i,j} U_i^T U_j\right] = 
\sum_{i,j = 1}^m L_{i,j} \operatorname{Tr}(U_i^T U_j) = 
\sum_{i,j = 1}^m L_{i,j} U_j U_i^T
$$

