Linear maps for which $f^{-1}=f$

Question

I could not find a particular name for linear maps which are their own inverse. Is there a special name for them?

Answer 1

Your use of the term linear maps and the linear-algebra tag suggest that you're asking about linear transformations, i.e., maps $T : V \to W$, where $V$ and $W$ are vector spaces.

Recall that every such linear transformation can be represented by some matrix $A$.

So asking if there is a $T : V \to W$ such that $T = T^{-1}$ is the same as asking if there's some matrix $A$ such that $A = A^{-1}$. Such matrices are known as involutory.

I'm not entirely sure if the word "involutory" is also used to describe $T$.

Answer 2

Those linear maps are involutions.