book 1, page 447, formula 13.69

[g-i-o-r-g-i-o]

I'm sorry, but I can't understand if the dimensions of the Jacobian and of the gradient are correct.

Issue:
The number of columns in J_l (nl) does not match the number of rows in g_l+1 (nl+1). This mismatch means that the matrix multiplication cannot be performed directly.

Assume the following layer dimensions:

Layer l: nl neurons
Layer l+1: nl+1 neurons

Given these, the shapes of the components are:

Gradient Vector at Layer l+1:
g_l+1 = dL / dz_l+1 
Shape: (nl+1, 1)

Jacobian Matrix J_l:
J_l = dz_l+1 / dz_l 
Shape: (nl+1, nl)

Gradient Vector at Layer l:
dL / dz_l 
Shape: (nl, 1)

The formula involves multiplying J_l with g_l+1:

J_l * g_l+1 
Shape: (nl+1, nl) * (nl+1, 1)