# Distinguishing between $\epsilon$ and $e$ in interaction with residual maker matrix $M$

I've hit a small snag in working out some of the implications of the residual maker matrix $$M$$.

Through previous posts I've been able to understand the difference between the use of $$e$$ and $$\epsilon$$, however there is still something I don't understand. We have the residual maker matrix, $$M = I-X(X'X)^{-1}X'$$, and we also know that $$e=My$$ since

$$e = y - \hat y = y - Py = y - X(X'X)^{-1}X'y = (I-X(X'X)^{-1}X')y = My$$

My concern, however, comes with when we evaluate this line further. We have two equivalencies for $$y$$. In the matrix form of our model we have $$y=X\beta + \epsilon$$. However, since we can also represent our residual as $$e=y-X\hat \beta$$ given that $$\hat y = X\hat \beta$$, we can also define $$y=X\hat \beta + e$$. If we substitute either $$y$$ into $$e= My$$, while also knowing that

$$MX=0$$ since $$(I-X(X'X)^{-1}X')X = X - X(X'X)^{-1}X'X = X- XI = X -X = 0$$,

we get that $$e$$ can be represented both as

$$e = My = M(X\hat \beta + e) = Me$$

and

$$e= My =M(X\beta + \epsilon) = M\epsilon$$.

If $$e=Me=M\epsilon$$, does that not necessitate that $$e=\epsilon$$, at least in this model? I've seen lecture notes write either one of these equivalencies but never both at the same time, so at this point I feel as if I'm running in circles. Any insight would be greatly appreciated!

The residual-maker matrix $$M$$ (also called the "hat matrix" and usually denoted as $$\mathbf{h}$$ or $$\mathbf{H}$$ in may sources) is a projection matrix that projects vectors in $$\mathbb{R}^n$$ onto the column-space of the design matrix. You are correct that it yields the following projection outputs:
$$M \epsilon = e \quad \quad \quad \quad \quad Me = e.$$
However, since the projection is not an injective operation, it does not follow from this that $$e = \epsilon$$. Different vectors can be projected onto the same vector by the projection. In this case we see that the error vector $$\epsilon$$ projects onto the residual vector $$e$$, and if we apply the projection to the residual vector $$e$$, nothing happens (since this vector is already in the column-space of the design matrix).