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Problem: Given $x=[1,7,2,3,-1]^T $and $y=[-4,4,4,0,-4]^T$, find an orthogonal matrix $Q$ such that $Qx=y$.

My attempt: I know the definition of an orthogonal matrix. By definition, if $Q$ is an orthogonal matrix then $Q^{-1}=Q^T$. Further, I know if system of equations $Ax=b$ where $A$ is square matrix, has a solution if $b$ is in column space of $A$ i.e. if $rank(A:b)=rank(A)$.

I didn't able to solve above problem :-( please help me.

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  • $\begingroup$ Look up Householder reflection. $\endgroup$ – Chris Custer Jan 1 at 7:04
  • $\begingroup$ Sir, I didn't know the notion of the Householder reflection. Can't we solve directly? $\endgroup$ – Akash Patalwanshi Jan 1 at 7:09
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If you let $v=\frac{x-y}{\mid\mid y-x\mid\mid}$.

$A=I-2vv^t$ should do the trick.

This is called the Householder reflection.

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  • $\begingroup$ Sir thanks for your answer. Sir your answer shows such matrix exists. but, Why such a matrix exists? and how we know when it doesn't exist? $\endgroup$ – Akash Patalwanshi Jan 1 at 7:35
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    $\begingroup$ Note that the two vectors have the same norm. $\endgroup$ – Chris Custer Jan 1 at 7:38
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    $\begingroup$ This shows that it works:math.stackexchange.com/a/1903322 $\endgroup$ – Chris Custer Jan 1 at 7:44

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