Implement Softmax Cross-Entropy Loss and Backprop (NumPy)

Problem: Implement Softmax Cross-Entropy (with Backprop) in NumPy Given a batch of logits Z (unnormalized scores) and labels y, implement in NumPy: Numerically stable softmax probabilities P. Mean cross-entropy loss L. The gradient of the loss w.r.t. logits dZ (no autograd). Input Z: float matrix of shape (N, C) y: int vector of shape (N,) with values in [0, C-1] Output loss: scalar (batch mean) dZ: array of shape (N, C) Constraints Must be numerically stable (e.g., subtract row-wise max). No automatic differentiation libraries. Time complexity should be O(N*C). Test cases Z=[[0,0]] , y=[0]: loss should be log(2), dZ=[[-0.5, 0.5]] Z=[[10,0,0]] , y=[0]: loss close to 0, gradient for class 0 is negative and near 0 For random Z, finite-difference gradient check should have small error (e.g., < 1e-5) Example Input Z=[[0,0]] y=[0] Output loss=0.6931471805599453 dZ=[[-0.5,0.5]]