$$ \mathbf{Ax = b} \\ \mathbf{A} \in \mathbb{R}^{m \times n}, \, \mathbf{x} \in \mathbb{R}^n, \, \mathbf{b} \in \mathbb{R}^m \\ m \geq n, \, k = \text{rank}(\mathbf{A}) \leq n $$
$$ \mathbf{r}(\mathbf{x}) = \mathbf{A}\mathbf{x} - \mathbf{b} \\ \mathcal{L}_p(\mathbf{x}) = \|\mathbf{r}(\mathbf{x})\|_2 + \lambda\|\mathbf{x}\|p \\ \mathbf{x}^* = \arg\min\mathbf{x} \mathcal{L}_p(\mathbf{x}) $$
$$ \text{Example: } m = 3, \, n = 2, \, k = 1 \\ \mathbf{A} \in \mathbb{R}^{3 \times 2}, \, \mathbf{x} \in \mathbb{R}^2, \, \mathbf{b} \in \mathbb{R}^3 \\[0.3cm]
\mathbf{A} = \begin{bmatrix} \uparrow & \uparrow \\ \mathbf{a}_1 & \alpha \mathbf{a}_1 \\ \downarrow & \downarrow \end{bmatrix}, \, \mathbf{x} = \begin{bmatrix} \chi_1 \\ \chi_2 \end{bmatrix} \\ \mathbf{Ax} = \chi_1 \mathbf{a}_1 + \chi_2 \alpha \mathbf{a}_1 = (\chi_1 + \alpha \chi_2) \mathbf{a}_1 $$
$$ \text{Case 1: } \mathbf{b} \in \text{col}(\mathbf{A}), \, \mathbf{b} = \beta \mathbf{a}_1 \\ \mathbf{r(x)} = (\chi_1 + \alpha \chi_2 - \beta) \mathbf{a}_1 \\[0.3cm]
\min_{\mathbf{x}} \|\mathbf{r}(\mathbf{x})\|2 = 0 \\ (\chi_1, \chi_2) = \argmin{\mathbf{x}} \|\mathbf{r(x)}\|_2 \\ \chi_2 = \frac{\beta - \chi_1}{\alpha}, \, \mathbf{a}_1 \neq \mathbf{0} $$
$$ \text{Case 2: } \mathbf{b} \notin \text{col}(\mathbf{A}), \, \mathbf{b} \neq \mathbf{0} \\ \mathbf{r(x)} = (\chi_1 + \alpha \chi_2) \mathbf{a}_1 - \mathbf{b} \\[0.3cm]
\min_{\mathbf{x}} \|\mathbf{r}(\mathbf{x})\|_2 > 0 \\ (\hat{\chi}_1, \hat{\chi}2) = \argmin{\mathbf{x}} \|\mathbf{r(x)}\|_2 \\ \hat{\chi}_2 = \ell(\hat{\chi}_1), \, \mathbf{a}_1 \neq \mathbf{0} $$
$$ \min_\mathbf{x} \lambda \|\mathbf{x}\|1 = c\text{min} \\ \argmin_\mathbf{x} \mathcal{L}_1(\mathbf{x}) = \begin{cases} (\chi_1^, 0) \\ (0, \chi_2^) \end{cases} \\[0.3cm]
\min_\mathbf{x} \lambda \|\mathbf{x}\|2 = d\text{min} \\ \argmin_\mathbf{x} \mathcal{L}_2(\mathbf{x}) = (\chi_1^, \chi_2^) $$
