[Physics] Representations of Pauli matrices involving outer product of qubit states

Question

Let $| 0 \rangle$ and $| 1 \rangle $ be the states of qubit. Let $\hat{\sigma_x}$, $\hat{\sigma_y}$, $\hat{\sigma_z}$ be Pauli matrices:
$$
\hat{\sigma}_{x} = \left( \begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array} \right), \;\;\;
\hat{\sigma}_{y} = \left( \begin{array}{cc} 0 & -i \\ i & 0 \end{array} \right), \;\;\;
\hat{\sigma}_{z} = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right).
$$
Using the representation $\hat{I} = | 0 \rangle \langle 0 | + |1\rangle \langle 1| $ I tried to receive the similar representation for Pauli matrices:
$$
\hat{\sigma}_x = | 1 \rangle \langle 0 | + |0\rangle \langle 1|, \\
\hat{\sigma}_y = -i| 1 \rangle \langle 0 | + i|0\rangle \langle 1|, \\
\hat{\sigma}_z = -| 0 \rangle \langle 0 | + |1\rangle \langle 1|, \\
$$
Is it correct? I'm not sure if it is possible but I assumed that $|0\rangle = (0,1)^{T}$ and $|1\rangle = (1,0)^{T}$ and I looked at $|0\rangle \langle 1|$ as on outer product $(0,1)^{T}(1,0)$. I'm afraid that I use matrices instead of operators etc. Help me please to understand the topic.

Answer 1

$\newcommand{\ket}[1]{\left|#1\right>}\newcommand{\bra}[1]{\left<#1\right|}$ You actually got the notation slightly wrong and had a typo in $I$, but you're almost right.

The usual convention are $$ \begin{align} \ket0&=\begin{bmatrix}1 \\ 0\end{bmatrix}& \ket1&=\begin{bmatrix}0 \\ 1\end{bmatrix}\\ \text{if }\ket\psi&=\begin{bmatrix}\alpha \\ \beta\end{bmatrix}& \text{then }\bra\psi&=\begin{bmatrix}\alpha^* & \beta^*\end{bmatrix} \end{align} $$ So we have $$ \begin{align} I&=\ket0\bra0+\ket1\bra1\\ \sigma_x&=\ket1\bra0+\ket0\bra1\\ \sigma_y&=i\ket1\bra0-i\ket0\bra1\\ \sigma_z&=\ket0\bra0-\ket1\bra1 \end{align} $$