# Why does the concept of a unitary matrix seem so similar to that of an invertible matrix?

I understand that the intuition behind a unitary operator is that it preserves the length of the vector it acts upon. Also $$U^\dagger U = I$$. Doesn't that just mean that $$U$$ is just an invertible operator which preserves distance? And that $$U^\dagger$$ is the inverse?

If not, where am I making my mistake in reasoning? How should I rewire my thinking to avoid this flawed paradigm?

If so, does that then mean that all distance preserving and invertible operators are also normal operators?

This is one of the definitions of a unitary $$-$$ a (bounded) linear surjective map that preserves distances (the inner product). From this it can be deduced that $$U$$ is invertible and $$U^{-1} = U^\dagger$$.
Unitaries are normal because $$U^\dagger U = U U^\dagger$$, clearly.