Let $C$ be a collection of all rank-1 $2\times 2$ matrices $\mathbf{A}=\mathbf{pq}^T$. I am trying to enumerate its subsets $C_1, C_2,....$ like
- $C_1$: Collection of all symmetric matrices with $a_{ii}=1$ and $\alpha\neq 0$
\begin{align} \mathbf{A}=\mathbf{pq}^T=\begin{bmatrix}1\\ \alpha\end{bmatrix}\begin{bmatrix}1& \alpha\end{bmatrix}=\begin{bmatrix}1& \alpha\\ \alpha& \alpha^2\end{bmatrix} \end{align}
$C_2$: Collection of all matrices such that $\|p\|_2=\|q\|_2=1$
$C_3$: Collection of all matrices with $a_{ii}=1$ and $\alpha \neq 0, \beta \neq 0$
\begin{align} \mathbf{A}=\mathbf{pq}^T=\begin{bmatrix}1\\ \alpha\end{bmatrix}\begin{bmatrix}1& \beta\end{bmatrix}=\begin{bmatrix}1& \beta\\ \alpha& \alpha\beta\end{bmatrix} \end{align}
Can someone help me in enumerating the rest.