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Advanced Proof

Orthogonality of Conjugate Directions

Students often conflate A A -conjugacy with Euclidean orthogonality. They are fundamentally different: while orthogonal vectors are perpendicular in the standard sense (piTpj=0 p_i^T p_j = 0 ), conjugate vectors are perpendicular only under the inner product defined by the operator A A , effectively warping the geometry to the function's curvature.
Institutional Reference: Fundamentals of Optimization
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