i have a problem in this question and can't solve it.
a) $0\ne w\in C^n$ is a column vector.find an neccesary and sufficient rule for w so the matrix $H=I-2ww^*$ will be unitary.
b) show that in this case H is a reflaction matrix to ${w}^\bot$(i.e show that Hw=-w and Hv=v for every $v \in {w}^\bot$.
what i did:
a)for a matrix to be unitary, it needs to follow $U^*U=UU^*=I$, so i tried to look for H so that $H^*H=HH^*=I$ where the column vector $w=\begin{pmatrix}z_1\\ ..\\ z_n\end{pmatrix}$ would become $w^*=(\overline{z_1},...,\overline{z_2})$, but couldn't find a definite and sufficient term for H to be unitary.
b)didn't know how to approach it since i couldn't prove that H is unitary.
don't know how to solve it, especially b.
thank you very much.