Projection matrix

The hat matrix, H, is used in statistics to relate errors in residuals to experimental errors. Suppose that a linear least squares problem is being addressed. The model can be written as

$\mathbf {y^{calc}=Jp}$

where J is a matrix of cefficients and p is a vector of parameters. The solution to the un-weighted least-squares equations is given by

$\mathbf {p=\left(J^{T}J\right)^{-1}J^{T}y^{obs}}$

The vector of un-wieghted residuals, r, is given by

$\mathbf {r=y^{obs}-y^{calc}=y^{obs}-J\left(J^{T}J\right)^{-1}J^{T}y^{obs}}$

The matrix $\mathbf {J\left(J^{T}J\right)^{-1}J^{T}}$ is known as the hat matrix. Thus the residuals can be expressed simply as

$\mathbf {r=\left(I-H\right)y^{obs}}$

The hat matrix is idempotent, that is, $\mathbf {HH=H}$. Using this property it can easily be shown by error propagation that the variance-covariance matrix of the residuals is equal to $\mathbf {I-H}$.Some other useful properties of the hat matrix are summarized in .