# Multiplication of a vector by an orthogonal matrix

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.

• I think WZ is a scalar, while U is a vector. So they cannot be equal. – Andrei Dec 6 '18 at 16:09
• no it's not a scalar, it's a vector – 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.
• 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 $u=Wz.$ – Adrian Keister Dec 6 '18 at 16:50