How can we convert the dot product of 3 vectors to a matrix form?
In particular, We have $\sum_{i=1}^m (\langle \vec{w},\vec{x_i}\rangle -y_i)\vec{x_i} = 0$, how and why can we convert it to the following matrix form?
$Aw = b$ where $A = (\sum_{i=1}^m x_i x_i^T)$ and $b = \sum_{i=1}^{m} y_i \vec{x_i}$
Detailed derivation will be welcome! Thank you!
First, $\sum_{i=1}^m \langle \vec{w},\vec{x_i}\rangle \vec{x_i}$ can be referred to the following form:
$(\vec{x_1}, \vec{x_2},\dots, \vec{x_m})$$\left[\begin{array}{c} \langle \vec{w},\vec{x_1}\rangle \\ \langle \vec{w},\vec{x_2}\rangle \\ \dots \\ \langle \vec{w},\vec{x_m}\rangle \\\end{array}\right] $
Then $\left[\begin{array}{c} \langle \vec{w},\vec{x_1}\rangle \\ \langle \vec{w},\vec{x_2}\rangle \\ \dots \\ \langle \vec{w},\vec{x_m}\rangle \\\end{array}\right] $ can be reformulated as
$\left[\begin{array}{c} \vec{x_1^T} \\ \vec{x_2^T} \\ \dots \\ \vec{x_m^T} \\\end{array}\right] $$w$
Then we can get
$(\sum_{i=1}^m x_i x_i^T) w$