```markdown
### Analytic Center of Linear Matrix Inequality

#### Objective
- **Minimize:** \(-\log \det X\)
- **Subject to:** \(\text{Tr}(A_i X) = b_i, \quad i = 1, \ldots, p\)

#### Variable
- \(X \in \mathbb{S}^n\)

#### Optimality Conditions
- \(X^\star \succ 0\)
- \(- (X^\star)^{-1} + \sum_{j=1}^{p} \nu_j^\star A_i = 0\)
- \(\text{Tr}(A_i X^\star) = b_i, \quad i = 1, \ldots, p\)

#### Newton Equation at Feasible \(X\)
- \(X^{-1} \Delta X X^{-1} + \sum_{j=1}^{p} w_j A_i = X^{-1}\)
- \(\text{Tr}(A_i \Delta X) = 0, \quad i = 1, \ldots, p\)

#### Notes
- Follows from linear approximation \((X + \Delta X)^{-1} \approx X^{-1} - X^{-1} \Delta X X^{-1}\)
- \(n(n+1)/2 + p\) variables \(\Delta X, w\)

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*Source: A. d’Aspremont. Convex Optimization M2.*
```
