3.2- Multi Neurons Matrix

Lets solve the same thing using matrix way

weight	input	bias
$Weights=\begin{bmatrix}neuron1=>\\neuron2=>\end{bmatrix}\begin{bmatrix} 1&2&3\\2&1&2\end{bmatrix}$	$inputs=\begin{bmatrix}2 \ 3 \ 4\end{bmatrix}$	$bias=\begin{bmatrix}1\ 2 \end{bmatrix}$
Substitute:

$$y= \begin{bmatrix} 1&2&3\\ 2&1&2 \end{bmatrix} \begin{bmatrix} 2\\ 3\\ 4 \end{bmatrix} + \begin{bmatrix} 1\\ 2 \end{bmatrix}$$

Matrix multiplication gives:

$$y= \begin{bmatrix} (1)(2)+(2)(3)+(3)(4)\ (2)(2)+(1)(3)+(2)(4) \end{bmatrix} + \begin{bmatrix} 1\ 2 \end{bmatrix}$$ $$y= \begin{bmatrix} 20\ 15 \end{bmatrix} + \begin{bmatrix} 1\ 2 \end{bmatrix}$$

Therefore:

$$\boxed{ y= \begin{bmatrix} 21\ 17 \end{bmatrix}}$$

---

The Important Connection

Expanded:

$$\begin{aligned} y_1&=w_{11}x_1+w_{12}x_2+w_{13}x_3+b_1\\ y_2&=w_{21}x_1+w_{22}x_2+w_{23}x_3+b_2 \end{aligned}$$

Contracted:

$$\boxed{y=Wx+b}$$

The matrix form is simply a compact way of writing all the individual neuron calculations together.

import numpy as np

x = np.array([2, 3, 4])

W = np.array([
    [1, 2, 3],
    [2, 1, 1]
])

b = np.array([5, 2])

y = W @ x + b

print(y)