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.

  • $\begingroup$ 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

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.

  • $\begingroup$ 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
  • 3
    $\begingroup$ Note that the two vectors have the same norm. $\endgroup$ – Chris Custer Jan 1 at 7:38
  • 1
    $\begingroup$ This shows that it works:math.stackexchange.com/a/1903322 $\endgroup$ – Chris Custer Jan 1 at 7:44

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.