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I am reading the lecture notes of EE263 of stanford university. I came across these 2 slides (Please see attached images).

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I understand every step up to the second slide where it says "residual with optimal x is".

My question is: why is $Ax_{ls} -y = -Q_2 Q^T_2y$ ? I do not know how this is derived. Can someone show me how this is derived ?

1 Answers 1

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It is a little confusing unless you see the trick:

The key here is $\begin{bmatrix} Q_1 Q_2\end{bmatrix} \begin{bmatrix} Q_1^T \\ Q_2^T\end{bmatrix} = Q_1 Q_1^T + Q_2 Q_2^T = I$.

Then $Ax_{ls} - y = Q_1 R_1 x_{ls} - y = Q_1 R_1 R_1^{-1}Q_1^T y - y = (Q_1 Q_1^T-I)y = - Q_2 Q_2^T y$.

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    why is $Q_1Q^T_1 + Q_2Q^T_2 = I$? – 2017-01-28
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    An orthogonal matrix $Q$ satisfies $Q Q^T = I$. The matrix $Q=[Q_1 Q_1]$ is orthogonal. – 2017-01-28
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    I meant $[Q_1 Q_2]$, of course. – 2017-01-28
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    you mean $Q=[Q_1 Q_2]$, right ? – 2017-01-28
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    Yes, I wrote $[Q_1 Q_1]$ by mistake last night. – 2017-01-28
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    yep yep. It's cool. I understood what you meant last nite. Thanks a lot man – 2017-01-28
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    Glad to help, I often trip on small points. Sometimes what is obvious at one moment is hidden by fog the next :-). That said, sometimes small and seemingly obvious results are laborious to prove. – 2017-01-28