TITLE: What's the interpretation of a unitary matrix?
QUESTION [19 upvotes]: I know that a unitary matrix is a matrix whose inverse equals its conjugate transpose (or that multiplying it by its conjugate transpose yields the identity), but I don't have a deep intuition about it (I just accept the definition). So for example, when I encounter the statement that the left and right singular decompositions $U$ and $V$ in the SVD are unitary, I don't get the significance. I would appreciate if somebody could enlighten me to connect the dots and how to feel when encountering unitary matrices. I have the feeling there is something unwritten that I'm missing.

REPLY [3 votes]: The unitary matrices are precisely those matrices which preserve the Hermitian inner product. 
$$\langle v,w \rangle = \langle Uv,Uw \rangle$$
Hence it preserves the distance or length of a vector in the Unitary space (the finite-dimensional vector space over the complex number field $\mathbb{C}$ with an inner product) under rotation or reflection.