Related question: this topic is examined
Least-Squares Problems
Least-squares solutions of inconsistent systems, the normal equations AᵀAx̂ = Aᵀb, uniqueness when the columns are independent, the least-squares error, and solving with QR.
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Least-squares solution x̂ minimizes ‖b − Ax‖. Solve the normal equations AᵀAx̂ = Aᵀb. Unique ⇔ A’s columns are independent ⇔ AᵀA invertible: x̂ = (AᵀA)⁻¹Aᵀb.
Tip Least-squares error = ‖b − Ax̂‖.
Exam angle
Related question: this topic is examined
Related question: this topic is examined
The least-squares solutions of Ax = b are the solutions of:
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Notes
With A = QR: Rx̂ = Qᵀb (more stable numerically).
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