# Find projection matrix using partitioned matrices

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 do you think I should prove the equation above? Will the formula I found on Wikipedia be helpful?

Thanks.

• Not kosher to cross-post math.stackexchange.com/questions/1830852/… . – Mark L. Stone Jun 18 '16 at 17:05
• You're going to have a hard time proving that a $p+1\times p+1$ matrix is the sum of a $1\times 1$ matrix and a $p\times p$ matrix! This suggests the first thing to clear up is the meaning of "$+$" in the formula you're trying to prove. – whuber Jun 18 '16 at 17:55