What is Ordinary least squares, hat matrix?

For $y \in \R^n$ and a full-rank design $X \in \R^{n \times k}$, ordinary least squares (OLS) chooses $\hat\beta = (X^\top X)^{-1}X^\top y$, the minimiser of $\lvert y - X\beta\rvert^2$. The fitted values are $\hat y = Hy$ with the hat matrix $H = X(X^\top X)^{-1}X^\top$, the orthogonal projection onto the column space of $X$; its diagonal entries $h_{ii}$, which sum to $k$, are the leverages of the observations.