I have a question, consider $V$ an orthogonal matrix, and $u$ and $z$ are vectors, and W is a matrix does :

$V'u = W V'z \implies u = W z$ ?

I want to get rid of the orthogonal matrix $V'$, my intuition says that I can, but I don't know which property of the orthogonal matrices will help me to do say. Thank you in advance.

  • 1
    $\begingroup$ I think WZ is a scalar, while U is a vector. So they cannot be equal. $\endgroup$ – Andrei Dec 6 '18 at 16:09
  • $\begingroup$ no it's not a scalar, it's a vector $\endgroup$ – math geek Dec 6 '18 at 16:12

Assuming $V'u = WV'z$, we have

$$V'(u - Wz) = (WV' - V'W) z.$$

For the left-hand side to be zero for arbitrary $z$, $W$ and $V'$ have to commute, so your statement is not true in general.

  • $\begingroup$ yes, my matrices do commute ! thank you for your answer, although i didn't fully understand why this equality is true : $\endgroup$ – math geek Dec 6 '18 at 16:46
  • $\begingroup$ this one V'(u - Wz) = (WV' - V'W) z. $\endgroup$ – math geek Dec 6 '18 at 16:46
  • $\begingroup$ Another way of thinking about it: if $V'$ and $W$ commute, then $V'u=WV'z$ if and only if $V'u=V'Wz$ if and only if ${V'}^{\dagger}V'u={V'}^{\dagger}V'Wz$ if and only if $u=Wz.$ $\endgroup$ – Adrian Keister Dec 6 '18 at 16:50

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