What is the hat matrix in Linear Regression and why is it important?

The hat matrix maps observed target values to predicted values and reveals influence of each data point.

The hat matrix H = X(XᵀX)⁻¹Xᵀ transforms y into predictions: ŷ = Hy. Its diagonal elements (hᵢᵢ) measure leverage of each point. It is fundamental in diagnostics because it quantifies how much each observation affects its own prediction.