```markdown
### Image Description

The image presents a slide on the KKT (Karush-Kuhn-Tucker) system from a lecture or presentation on convex optimization.

#### Content:

- **Title**: KKT system
- **Matrix Equation**:
  \[
  \begin{bmatrix} H & A^T \\ A & 0 \end{bmatrix} \begin{bmatrix} v \\ w \end{bmatrix} = - \begin{bmatrix} g \\ h \end{bmatrix}
  \]
- **Details**:
  - \( H = \text{diag}(\phi''_1(x_1), \ldots, \phi''_n(x_n)) \), indicating a positive diagonal matrix.
  - The system can be solved via elimination:
    \[
    AH^{-1}A^Tw = h - AH^{-1}g, \quad Hv = -(g + A^Tw)
    \]
- **Sparsity Pattern**:
  - The sparsity pattern of the coefficient matrix is given by graph connectivity.
  - Conditions:
    \((AH^{-1}A^T)_{ij} \neq 0 \iff (AA^T)_{ij} \neq 0\)
  - This implies nodes \(i\) and \(j\) are connected by an arc.

- **Footer**: A. d'Aspremont. Convex Optimization M2. Page 50/53.
```
