I have a particular regression model where my design matrix $X$ has the property that each row sums to $1$ and are all positive. I'm curious if this implies anything about the hat matrix $H$.
$$H = X(X^TX)^{-1}X^T$$
Does this imply that the hat matrix diagonals are <1? Or something about positive definite? It seems like there must be some structure of $H$ under such a condition?
The matrix $H$ is an orthogonal projector, that is, $H^T=H$ and $H^2=H$.
The two conditions imply that $H$ is positive semidefinite because $$ x^THx=x^TH^2x=x^TH^THx=(Hx)^T(Hx)\geq 0\quad\forall x. $$ This implies that the diagonal entries of $H$ are non-negative by substituting $x=e_i$ where $e_i$ is the $i$th column of identity.
We also have that $$ e_i^THe_i\leq\max_{\|x\|_2=1}x^THx=\lambda_{\max}(H)\leq\|H\|_2. $$ Because $H$ is idempotent, $\|H\|_2=\|H^2\|_2\leq\|H\|_2^2$ and hence $\|H\|_2\leq 1$ (actually, for non-trivial orthogonal projectors, $\|H\|_2=1$. Consequently, all diagonals elements $e_i^THe_i$ of $H$ are in the interval $[0,1]$.
Note that this has nothing to do with properties of $X$.
