In a neural network, matrix multiplication is used to compute the output of a layer by multiplying the input matrix by the weight matrix of that layer. The weights represent the learned parameters of the neural network, and the matrix multiplication essentially combines the input data with these learned weights to produce the output. After each matrix multiplication, a nonlinear activation function is typically applied to introduce nonlinearity and increase the expressiveness of the model.

During training, matrix multiplication is used to compute the gradients of the loss function with respect to the weights. This involves multiplying the error signal by the transpose of the input matrix to compute the gradient of the weights, which is then used to update the weights in the direction that minimizes the loss.

Efficient implementations of matrix multiplication are critical for training deep neural networks, which can involve millions or even billions of parameters. Researchers have developed highly optimized algorithms for matrix multiplication, including those that can take advantage of specialized hardware like GPUs and TPUs, to speed up the training process and improve the performance of deep learning models.