Linear Algebra and Its Applications · Problem 3.3.41

Orthogonality — Gilbert Strang

Mathematics · weighted-least-squares

Problem

Find the weighted least-squares solution x^W\hat{x}_W to Ax=bAx = b: A=[101112]b=[011]W=[200010001].A = \begin{bmatrix} 1 & 0 \\ 1 & 1 \\ 1 & 2 \end{bmatrix} \quad b = \begin{bmatrix} 0 \\ 1 \\ 1 \end{bmatrix} \quad W = \begin{bmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}. Check that the projection Ax^WA\hat{x}_W is still perpendicular (in the WW-inner product!) to the error b−Ax^Wb - A\hat{x}_W.

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