If X is a ($n$, $p+1$) design matrix, partition $X$ to be $X$=[$J$ $X$*] where $J$ is a ($n$,$1$) vector of all $1$'s, and $X$* is a ($n$,$p$) matrix.
Let $H_X$ be a projection matrix, where $H_X$ = $X{(X'X)}^{-1}X'$.
I'm trying to prove:
$H_X$ = $H_J$ + $H_X*$
And in doing so, I've been trying to deduce the above equation using:
Suppose the design matrix $X$ can be decomposed by columns as $X$= [$A$ $B$]. Define the hat or projection operator as $P${$X$} = $X{(X'X)}^{-1}X'$.
Similarly, define the residual operator as $M${$X$}=$I$-$P${$X$}. Then the projection matrix can be decomposed as follows:$P${$X$} = $P${$A$} + $P${$M${$A$}$B$}.
I found this formula (the "Blockwise Formula") on Wikipedia. The link is: https://en.wikipedia.org/wiki/Projection_matrix
I've been trying to prove this formula, and from that deduce the equation I'm trying to solve. But I still haven't found a way to prove either.
How can I prove the equation above?
Will the formula I found on Wikipedia be helpful?
Here is a counter-example: $$\begin{bmatrix}1&1&0&0\\1&0&1&0\\1&0&0&1\\1&0&0&0 \end{bmatrix}.$$ You can check that in this case $H_X=I_4 \neq H_J + H_{X^*}$.