# Your solution is here
## Problem 1
$$
\frac{1}{2} ||X - UV||^2_F =
\frac{1}{2} \langle X - UV, X - UV \rangle =
\frac{1}{2} (\langle X, X \rangle - 2 \langle X, UV \rangle + \langle UV, UV \rangle)
$$

### U derivative
$$
d F(U, V) =
\frac{1}{2} (2 \langle d(UV), UV) \rangle - 2 \langle d(UV), X \rangle =
\langle UV, dU V \rangle - \langle X, dU V \rangle =
\langle UVV^* - XV^*, dU \rangle
$$

Then, $grad F(U, V)$ w.r.t. U equals $(UV - X)V^*$

### V derivative
Accordingly, we get gradient w.r.t. V: $grad F(U, V) = U^*(UV - X)$

### Complexity
We multiply $n \times k$ matrix with $k \times n$, what gives us $O(n^2 k)$ complexity. Then we have a sum of 2 matrices, which has $O(n^2)$ complexity and finally we have muliplication with complexity $O(n^2 k)$. Summing up results, we have complexity of $O(n^2 k + n^2 + n^2k) = O(n^2 k)$

## Problem 2
$$
d R(x) =
\frac{\langle x, x \rangle d\langle Ax, x \rangle - \langle Ax, x \rangle d \langle x, x \rangle}{\langle x, x \rangle^2} =
\frac{\langle x, x \rangle \langle (A + A^T) x, dx \rangle - \langle Ax, x \rangle \langle x, dx \rangle}{\langle x, x \rangle^2} =
\frac{\langle \langle x, x \rangle (A + A^T)x - \langle Ax, x \rangle x\rangle, dx}{\langle x, x \rangle^2}
$$

## Problem 3
Let's denote $\sum_{i = 1}^{m} w_i x_i x_i^T$ as $S$.

Then we have:
$$
d f(w) = \frac{\det S * \langle S^{-T}, dS \rangle}{\det S} =
\langle S^{-T}, dS \rangle
$$

Let's find derivative of $S$ w.r.t. $w_i$:
$$
\frac{\partial S}{\partial w_i} = x_i x_i^T
$$

Then we can derive a coordinate of the gradient we are looking for:
$$
\frac{\partial f(w)}{\partial w_i} = \langle \left( \sum_{i = 1}^{m} w_i x_i x_i^T \right)^{-T}, x_i x_i^T \rangle
$$