Prove that there is a symmetric matrix B, such that BX=Y Let $X,Y$ be two vectors in ${\mathbb C}^n$ and assume that $X≠0$. Prove that there is a symmetric matrix $B$ such that $BX=Y$.

Hint 1: If $X$ has no zero entry, prove that you can find a diagonal $B$ so that $BX=Y$.
Hint 2: If $X$ has some zero entries, since $X \neq 0$ prove that you can find some orthogonal matrix $P$ so that $PX$ has no zero entry.
Define $X'=PX$ and $Y'=PY$. Find a diagonal matrix $B'$ so that $B'X'=Y'$ and use 
$$
B=P^{-1}B'P 
$$
