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# Sinkhorn's theorem

Sinkhorn's theorem states that every square matrix with positive entries can be written in a certain standard form.

Theorem

If *A* is an *n* × *n* matrix with strictly positive elements, then there exist diagonal matrices *D*_{1} and *D*_{2} with strictly positive diagonal elements such that *D*_{1}*AD*_{2} is doubly stochastic. The matrices *D*_{1} and *D*_{2} are unique modulo multiplying the first matrix by a positive number and dividing the second one by the same number. ^{[1]} ^{[2]}

Sinkhorn-Knopp algorithm

A simple iterative method to approach the double stochastic matrix is to alternately rescale all rows and all columns of A to sum to 1. Sinkhorn and Knopp presented this algorithm and analyzed its convergence. [3]

Analogues

The following analogue for unitary matrices is also true: for every unitary matrix U there exist two diagonal unitary matrices L and R such that LUR has each of its columns and rows summing to 1.[4]

References

Sinkhorn, Richard. (1964). "A relationship between arbitrary positive matrices and doubly stochastic matrices." Ann. Math. Statist. 35, 876–879. doi:10.1214/aoms/1177703591

Marshall, A.W., & Olkin, I. (1967). "Scaling of matrices to achieve specified row and column sums." Numerische Mathematik. 12(1), 83–90. doi:10.1007/BF02170999

Sinkhorn, Richard, & Knopp, Paul. (1967). "Concerning nonnegative matrices and doubly stochastic matrices". Pacific J. Math. 21, 343–348.

Idel, Martin; Wolf, Michael M. (2015). "Sinkhorn normal form for unitary matrices". Linear Algebra and its Applications 471: 76–84. doi:10.1016/j.laa.2014.12.031.

Quadratic forms with the same core form are said to be similar or Witt equivalent.

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