User:Nordron/sandbox

SVD of a Permuted Matrix
The singular values of a matrix are invariant under row/column interchanges (permutations) and the new $U$ and $V$ matrices are permuted in a corresponding manner. This can be established by considering the permuted matrix
 * $$\tilde{\mathbf{M}} = \mathbf{P} \mathbf{M} \mathbf{Q}^*

= \tilde{\mathbf{U}} \tilde{\boldsymbol{\Sigma}} \tilde{\mathbf{V}}^*$$

where $P$ and $Q$ are permutation matrices. On substituting the SVD of $M$, we have


 * $$\tilde{\mathbf{M}} = \mathbf{P} \mathbf{U} {\boldsymbol{\Sigma}} \mathbf{V}^* \mathbf{Q}^*

= \tilde{\mathbf{U}} \tilde{\boldsymbol{\Sigma}} \tilde{\mathbf{V}}^* $$

from which it follows that


 * $$\tilde{\boldsymbol{\Sigma}} = {\boldsymbol{\Sigma}} \text{, } \tilde{\mathbf{U}} = \mathbf{P} \mathbf{U} \text{, and }

\tilde{\mathbf{V}} = \mathbf{Q} \mathbf{V} $$

where $$ \tilde{\mathbf{U}} $$ and $$ \tilde{\mathbf{V}} $$ are unitary matrices as required.

The conjugate transpose of $M$ can be treated in a similar manner by letting


 * $$\tilde{\mathbf{M}} = {\mathbf{M}}^* = {\mathbf{V}} {\boldsymbol{\Sigma}} {\mathbf{U}}^*

= \tilde{\mathbf{U}} \tilde{\boldsymbol{\Sigma}} \tilde{\mathbf{V}}^* $$

whence


 * $$\tilde{\boldsymbol{\Sigma}} = {\boldsymbol{\Sigma}} \text{, } \tilde{\mathbf{U}} = \mathbf{V} \text{, and }

\tilde{\mathbf{V}} = \mathbf{U} $$

The conjugate transpose and row/column permutation can be performed in either order.