Is there a name for the matrix $X(X^tX)^{-1}X^{t}$? In my work, I have repeatedly stumbled across the matrix (with a generic matrix $X$ of dimensions $m\times n$ with $m>n$ given) $\Lambda=X(X^tX)^{-1}X^{t}$. It can be characterized by the following:
(1) If $v$ is in the span of the column vectors of $X$, then $\Lambda v=v$.
(2) If $v$ is orthogonal to the span of the column vectors of $X$, then $\Lambda v = 0$.
(we assume that $X$ has full rank).
I find this matrix neat, but for my work (in statistics) I need more intuition behind it. What does it mean in a probability context? We are deriving properties of linear regressions, where each row in $X$ is an observation.
Is this matrix known, and if so in what context (statistics would be optimal but if it is a celebrated operation in differential geometry, I'd be curious to hear as well)?

This should be a comment, but I can't leave comments yet.  As pointed out by Rahul Narain, this is the orthogonal projection onto the column space of $X$
