If Q is a matrix of orthonormal bases, is Q times its transpose anything special? I know that if Q is a matrix of orthonormal bases then Q transpose times Q is the identity matrix. Is Q times Q transpose anything special?

It is also the identity matrix. This holds in general: if $B$ is a left inverse of a square matrix $A$, i.e. $BA=I$, then $B$ is also a right-inverse of $A$, i.e. $AB=I$.
