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### Solution by Block Elimination

- **Eliminate** \( \Delta X \) from the first equation: 
  \[
  \Delta X = X - \sum_{j=1}^{p} w_j X A_j X
  \]

- **Substitute** \( \Delta X \) in the second equation:
  \[
  \sum_{j=1}^{p} \text{Tr}(A_i X A_j X) w_j = b_i, \quad i = 1, \ldots, p \quad (2)
  \]
  
  A dense positive definite set of linear equations with variable \( w \in \mathbb{R}^p \).

### Flop Count (Dominant Terms) Using Cholesky Factorization \( X = LL^T \):

- Form \( p \) products \( L^T A_j L \): \((3/2)pn^3\)
- Form \( p(p + 1)/2 \) inner products \(\text{Tr}((L^T A_i L)(L^T A_j L))\): \((1/2)p^2 n^2\)
- Solve (2) via Cholesky factorization: \((1/3)p^3\)

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*Source: A. d’Aspremont. Convex Optimization M2. Slide 52/53*
```