Prove H and I-H are Idempotent

Define the hat matrix \(H=X(X^TX)^{-1}X^T\).

For H to be Idempotent then \(HH=H\)

\[\begin{equation}\label{HH=H} \begin{split} HH & =[X(X^TX)^{-1}X^T][X(X^TX)^{-1}X^T]\\ & = X(X^TX)^{-1}X^TX(X^TX)^{-1}X^T\quad\quad(X^TX)^{-1}X^TX=1\\ & = X(X^TX)^{-1}X^T\\ & = H \end{split} \end{equation}\]

Therefore by the series of equalities H is idempotent.

For I-H to be idempotent then \((I-H)(I-H)=I-H\)

\[\begin{equation}\label{I-H} \begin{split} (I-H)(I-H) & =II-HI-IH+HH\quad\quad II=I, HI=IH=H, HH=H\\ & = I-H-H+H\\ & = I-H \end{split} \end{equation}\]

Therefor by the series of equalities I-H is idempotent.

QED.