Explain numerical stability concerns with matrix inversion in Kalman filter

[Copilot]

Addressed review question about why np.linalg.inv(S) is considered numerically unstable in the Kalman gain computation.

Explanation provided

The current implementation on line 62 of kf_emulator.py:

K = Sigma_pred @ self.H.T @ np.linalg.inv(S)

Computing explicit matrix inverse amplifies numerical errors for ill-conditioned matrices. In Kalman filtering, the innovation covariance S can become ill-conditioned when observation noise is small or predicted covariance has disparate magnitudes across dimensions.

Using np.linalg.solve(S, rhs) instead:

• Solves the linear system directly via LU decomposition without computing the inverse
• Reduces floating-point operations by ~33%
• Produces more accurate results when S is nearly singular

The mathematically equivalent but more stable form:

K = np.linalg.solve(S, (Sigma_pred @ self.H.T).T).T

No code changes made in this PR—purely explanatory response to review comment.

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