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arXiv · 1408.5728

Sinkhorn normal form for unitary matrices

Martin Idel·Michael M. Wolf
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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.

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Martin Idel, Michael M. Wolf. 2014-08-25. Sinkhorn normal form for unitary matrices. https://doi.org/10.1016/j.laa.2014.12.031

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