3.1- Multiple Neurons — Vector and Matrix Form

Suppose we have 2 neurons and 3 inputs. Each neuron receives the same inputs, but has its own weights and bias.

Neuron 1	Neuron 2
$\boxed{y_1=w_{11}x_1+w_{12}x_2+w_{13}x_3+b_1}$	$\boxed{y_2=w_{21}x_1+w_{22}x_2+w_{23}x_3+b_2}$

Vector Form

Put the two outputs together:

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

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Matrix Form

Collect all the weights into a matrix:

$$W= \begin{bmatrix} w_{11}&w_{12}&w_{13}\\ w_{21}&w_{22}&w_{23} \end{bmatrix}$$

Inputs:

$$x= \begin{bmatrix} x_1\\ x_2\\ x_3 \end{bmatrix}$$

Biases:

$$b= \begin{bmatrix} b_1\\ b_2 \end{bmatrix}$$

Expanded:

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

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Example

Suppose we have 2 neurons and 3 inputs.

Inputs and Parameters

	weight 1	weight 2	weight 3	bias
Neuron 1	1	2	3	1
Neuron 2	2	1	2	2

input	x1	x2	x3
value	2	3	4

Expanded Form

Neuron 1

Neuron	Neuron 1
1	$y_1=w_{11}x_1+w_{12}x_2+w_{13}x_3+b_1$	$y_2=w_{21}x_1+w_{22}x_2+w_{23}x_3+b_2$
2	$y_1=(1)(2)+(2)(3)+(3)(4)+1$	$y_2=(2)(2)+(1)(3)+(2)(4)+2$
3	$\boxed{y_1=21}$	$\boxed{y_2=17}$
Therefore:

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

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