How to derivate norms?

Question

I would like to solve this formula:

$$ \frac{\partial}{\partial v} \frac{1}{2} \left( \| A*v + c \|_2^2 + \|v\|_F^2 \right) $$

Where: $ A \in \mathbb{R}^{n \times n} $, and $ v \in \mathbb{R}^n$

Without the norms it is easy, but I have no idea, how to handle the Frobenius, and 2 norm.

Answer 1

Convert it to a form that you are more familiar without norm is an option. \begin{align} &\frac{\partial}{\partial v} \frac{1}{2} \left( \| Av + c \|_2^2 + \|v\|_F^2 \right) \\ &=\frac{\partial}{\partial v} \frac{1}{2} \left( ( Av + c)^T(Av+c) + \|v\|_2^2 \right) \\ &= \frac{\partial}{\partial v} \frac{1}{2} \left( (v^TA^TAv+c^TAv+v^TA^Tc + v^Tv \right) \\ &= \frac{1}{2} \left( (2A^TAv+A^Tc+A^Tc + 2v \right) \\ &= A^TAv+A^Tc+ v \\ &= A^T(Av+c)+ v \\ \end{align}