In the context of Machine Learning, how does matrix multiplication facilitate the process of gradient descent when optimizing a model's parameters?

Gradient descent updates a model’s parameters by moving them in the direction that most reduces the error, and this direction is found by computing gradients. In machine learning, these gradients are expressed as products of matrices that represent data, weights, and error signals; matrix multiplication lets us calculate all parameter updates at once instead of one at a time. For example, when training a linear regression with a feature matrix X and weight vector w, the gradient is the product Xᵀ(Xw−y), where Xᵀ(Xw−y) is obtained by multiplying Xᵀ with the error vector (Xw−y). This single matrix multiplication yields the gradient for every weight simultaneously, making the update step efficient and scalable. Thus, matrix multiplication turns a complex, high‑dimensional optimization problem into a series of fast, vectorized operations that drive the parameters toward the best fit.