Find a formula for the least-squares solution of Ax = b when the columns of A are orthonormal.
2026-03-29 13:47:17.1774792037
Finding the least-squares solution of Ax = b when the columns of A are orthonormal.
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We can write $RSS(x) = (\mathbf{b} - \mathbf{A}x)^T(\mathbf{b} - \mathbf{A}x)$. Differenciating w.r.t. $x$ we get the normal equations: $\mathbf{A}^T\left(\mathbf{b} - \mathbf{A}x\right) = 0$.
We know that $\mathbf{A}^T\mathbf{A} = \mathbb{I}$. Then $x = \mathbf{A}^T\mathbf{b}.$