I' ve been reading the review Ulrich Schollwöck: The density-matrix renormalization group in the age of matrix product states (arXiv link) and encountered with a question about the so called 'reduced basis transformation' on page 125 (journal page number; page 39 in the arXiv preprint).
To summarize, C is a square matrix with SVD decomposition $C=U^\dagger SV$，$C^\prime$ is another matrix that can be written as \begin{equation} C^\prime=P^\dagger \left [ \begin{matrix} S& 0&0 \\ 0& S &0\\ 0&0&S \end{matrix} \right]Q \end{equation} where $P^\dagger P=I$ and $Q^\dagger Q=I$, but $P P^\dagger,QQ^\dagger\neq I$, it is said that the largest singular value of $C^\prime$ is smaller than that of C, because this transformation corresponds to a reduced basis transformation into orthogonal subspaces.
My question is what is reduced basis transformation and how is it related to the singular value.