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- Jan 26, 2012

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**Problem**: Through transformation with orthogonal matrix $O$, the problem \(\displaystyle \hat{b}=\underset{b}{\operatorname{arg min}}||y-Xb||^2\) is equivalent to \(\displaystyle \hat{b}=\underset{b}{\operatorname{arg min}}||y^{*}-X^{*}b||^2\), where $y$ and $y^{*}$ are in $\mathbb{R}^m$, $X$ and $X^{*}$ are in $\mathbb{R}^{m \times n}$ ($m \ge n$) and $y^{*}=Oy$ and $X^{*}=OX$. Let $y^{*}=[y_1^{*},y_2^{*}...,y_m^{*}]^T$.

Prove that the residual sum of square \(\displaystyle ||y-X \hat{b}||^2=\sum_{i=n+1}^{m}||y_i^{*}||^2\).

**Solution**: I must admit I am a bit overwhelmed by this problem. I believe that $\underset{b}{\operatorname{arg min}}$ just means "for the lowest value of $b$", correct? I think I should start by reading up on the concepts which are needed to solve this. Can someone highlight the main ideas I'll need to know to attack this?