Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4599129 | Linear Algebra and its Applications | 2015 | 9 Pages |
Abstract
Sinkhorn proved that every entry-wise positive matrix can be made doubly stochastic by multiplying with two diagonal matrices. In this note we prove a recently conjectured analogue for unitary matrices: every unitary can be decomposed into two diagonal unitaries and one whose row- and column sums are equal to one. The proof is non-constructive and based on a reformulation in terms of symplectic topology. As a corollary, we obtain a decomposition of unitary matrices into an interlaced product of unitary diagonal matrices and discrete Fourier transformations. This provides a new decomposition of linear optics arrays into phase shifters and canonical multiports described by Fourier transformations.
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Martin Idel, Michael M. Wolf,