Numerische Mathematik

, Volume 12, Issue 1, pp 83–90 | Cite as

Scaling of matrices to achieve specified row and column sums

  • Albert W. Marshall
  • Ingram Olkin


IfA is ann ×n matrix with strictly positive elements, then according to a theorem ofSinkhorn, there exist diagonal matricesD1 andD2 with strictly positive diagonal elements such thatD1A D2 is doubly stochastic. This note offers an alternative proof of a generalization due toBrualdi, Parter andScheider, and independently toSinkhorn andKnopp, who show that A need not be strictly positive, but only fully indecomposable. In addition, we show that the same scaling is possible (withD1 =D2) whenA is strictly copositive, and also discuss related scaling for rectangular matrices. The proofs given show thatD1 andD2 can be obtained as the solution of an appropriate extremal problem.

The scaled matrixD1A D2 is of interest in connection with the problem of estimating the transition matrix of a Markov chain which is known to be doubly stochastic. The scaling may also be of interest as an aid in numerical computations.


Markov Chain Numerical Computation Mathematical Method Diagonal Element Transition Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 1968

Authors and Affiliations

  • Albert W. Marshall
    • 1
  • Ingram Olkin
    • 2
  1. 1.Boeing Scientific Research LaboratoriesSeattle
  2. 2.Department of Statistics Stanford UniversityStanford

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