Special Quantum Gate

Question

Given $\alpha|1\rangle+\beta|0\rangle$, to transform to $|\phi\rangle=\alpha|0\rangle+\beta|1\rangle$ , we use Quantum NOT gate:

NOT = $\begin{pmatrix}0 & 1 \\ 1 & 0\end{pmatrix}$.

For $\alpha|0\rangle-\beta|1\rangle$, what kind of gate do we need to have to transform to $|\phi\rangle$ ? Is it ok that we construct a gate as:

A = $\begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$ ?

Answer 1

I'm assuming you're representing $|0\rangle \simeq \binom{1}{0}$ and $|1\rangle \simeq \binom{0}{1}$. If you are, then you would be correct.