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I am facing the following problem in the context of linear regression:

On the one hand, it holds that $SS_{reg}=\sum(\hat{Yi} - \bar{Y})^2= Y'(H - \frac 1 nJ)Y$ where $H = X(X'X)^{-1}X'$ and $J$ is a matrix of ones (see e.g. https://math.stackexchange.com/q/1334601). If I understand this equality correctly this should imply that $(H - \frac 1 nJ)$ is idempotent. That is because $SS_{reg}=\sum(\hat{Yi} - \bar{Y})^2 = (\hat{Y} - \bar{Y})'(\hat{Y} - \bar{Y}) = ((H - \frac 1 n J)Y)'((H - \frac 1 n J)Y) = Y'(H - \frac 1 n J)(H - \frac 1 n J)Y$.

If I compute the product numerically the results also suggest that $H - \frac 1 n J$ is idempotent.

However, $(H - \frac 1 n J)(H - \frac 1 n J) = H - \frac 1 nHJ-\frac 1 n JH+\frac 1 n J$ which would suggest that $(H - \frac 1 n J)$ is not idempotent because $H$ is of course not the identity (see e.g. Proving that $\mathbf{(H-\frac{1}{n}J_n)}$ is indempotent). I don't know what I am missing.

jonaden
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    If you know that $H$ is the projection into the column space of $X$, and also have a column of 1’s in $X$, then it should be easy to argue that $H J / n = J / n$. The conceptual mistake you appear to be making is assuming that $AB = B$ (for some $B$) implies $A$ is the identity. – guy Dec 03 '20 at 18:01
  • Answered at https://math.stackexchange.com/a/1427733/321264. – StubbornAtom Apr 19 '22 at 20:37

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