I am confused about parts of the following excerpt from a textbook I am studying:
Degenerate conics. If the matrix $C$ is not of full rank, then the conic is termed degenerate. Degenerate point conics include two lines (rank 2), and a repeated line (rank 1).
Example. The conic
$$C = \mathbf{l}\mathbf{m}^T + \mathbf{m} \mathbf{l}^T$$
is composed of two lines $\mathbf{l}$ and $\mathbf{m}$. Points on $\mathbf{l}$ satisfy $\mathbf{l}^T \mathbf{x} = 0$, and are on the conic since $\mathbf{x}^T C \mathbf{x} = (\mathbf{x}^T \mathbf{l})(\mathbf{m}^T \mathbf{x}) + (\mathbf{x}^T \mathbf{m})(\mathbf{l}^T \mathbf{x}) = 0$. Similarly, points satisfying $\mathbf{m}^T \mathbf{x} = 0$ also satisfy $\mathbf{x}^T C \mathbf{x} = 0$. The matrix $C$ is symmetric and has rank 2. The null vector is $\mathbf{x} = \mathbf{l} \times \mathbf{m}$ which is the intersection point of $\mathbf{l}$ and $\mathbf{m}$.
Degenerate line conics include two points (rank 2), and a repeated point (rank 1). For example, the line conic $C^* = \mathbf{x} \mathbf{y}^T + \mathbf{y} \mathbf{x}^T$ has rank 2 and consists of lines passing through either of the two points $\mathbf{x}$ and $\mathbf{y}$. Note that for matrices that are not invertible $(C^*)^* \not= C$.
The questions I have are:
- How come $C = \mathbf{l}\mathbf{m}^T + \mathbf{m} \mathbf{l}^T$ results in a degenerate conic? How did the authors come up with the idea to define $C$ in such a way? In the earlier parts of the textbook, $C$ is simply defined as a $3\times3$ matrix consisting of the coefficients of the conic in quadratic form. I don't get why the sum of the vector product conveniently results in a conic matrix of such form.
- What is the significant of the null vector of $C$? Why is the conclusion that $\mathbf{x} = \mathbf{l} \times \mathbf{m}$ of an importance?