If Q is an orthogonal matrix that has orthonormal vectors as columns, I can total understand the following result $$Q^TQ=I$$ but $QQ^T=I$ only if Q is a square matrix and not if Q is a rectangular matrix. I cannot understand why the second result is true. When I'm doing $QQ^T=I$ I'm not taking dot product of orthonormal vectors and yet it yields as identity matrix for a square matrix. How is this possible?
Edit: Orthonormal changed to Orthogonal