Inverse of a reflection matrix?

Question

What does it mean when I say that the inverse of a reflection matrix is the reflection matrix itself? What does it mean, intuitively or geometrically, to invert a reflection matrix? Explanation with an example would help.

Answer 1

I think the type of reflection you're alluding to is a reflection through a hyperplane. Each point is sent to is "mirror image" on the other side of the hyperplane.

A better way to describe it is that the original point and its mirror image exchange places. That's why when the transformation is applied twice, everything returns to its original position, and so applying a reflection twice results in the identity transformation.

Inverting a reflection matrix is no different than inverting any other nonsingular matrix. The inverse undoes whatever the original transformation does.

Answer 2

The essence of any kind of reflection is that it is an involution of some space $X$, i.e., map $\iota:\>X\to X$ which is not the identity, but its square $\iota\circ\iota$ is the identity: Applying the same reflection a second time carries every point to the place where it originally was. But this is exactly saying that $\iota^{-1}=\iota$.

Now if you have a reflection $\iota$ that appears as a linear map then $\iota^{-1}=\iota$ imediately implies $A^{-1}=A$ for the matrix $A$ describing $\iota$, whatever the chosen basis is.