Skip to main content
SpringerLink
Log in

Doubly stochastic matrix equations

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

It is shown that for real,m x n matricesA andB the system of matrix equationsAX=B, BY=A is solvable forX andY doubly stochastic if and only ifA=BP for some permutation matrixP. This result is then used to derive other equations and to characterize the Green’s relations on the semigroup Ω n of alln x n doubly stochastic matrices. The regular matrices in Ω n are characterized in several ways by use of the Moore-Penrose generalized inverse. It is shown that a regular matrix in Ω n is orthostochastic and that it is unitarily similar to a diagnonal matrix if and only if it belongs to a subgroup of Ω n . The paper is concluded with extensions of some of these results to the convex setS n of alln x n nonnegative matrices having row and column sums at most one.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Ben-Israel and A. Charnes,Contributions to the theory of generalized inverses, J. Soc. Indust. Appl. Math.XI (1963), 667–699.

    Article  MathSciNet  Google Scholar 

  2. A. Berman and R. Plemmons,Monotonicity and the generalized inverse, SIAM J. Appl. Math.22, (1972), 155–161.

    Article  MathSciNet  Google Scholar 

  3. T. L. Boullion and P. L. Odell,Generalized Inverse Matrices, John Wiley and Sons, New York, 1971.

    MATH  Google Scholar 

  4. A. H. Clifford and G. B. Preston,The Algebraic Theory of Semigroups, Amer. Math. Soc. Surveys, Vol. I, 1967.

  5. J. L. Doob,Topics in the theory of Markov chains, Trans. Amer. Math. Soc.52, (1942), 37–64.

    Article  MATH  MathSciNet  Google Scholar 

  6. I. Erdelyi,On partial isometries in finite dimensional Euclidean spaces, J. Soc. Indust. Appl. Math.XIV (1966), 453–467.

    Article  MathSciNet  Google Scholar 

  7. H. K. Farahat,The semigroup of doubly stochastic matrices, Proc. Glasgow Math. Assoc.7 (1966), 178–183.

    MATH  MathSciNet  Google Scholar 

  8. T. N. E. Greville,Some applications of the pseudoinverse of a matrix, SIAM Rev.II (1960), 15–22.

    Article  MathSciNet  Google Scholar 

  9. G. H. Hardy, J. E. Littlewood and G. Pólya,Inequalities, Cambridge Press, Cambridge, England, 1934.

    Google Scholar 

  10. D. J. Hartfiel,The matrix equation AXB=X, Pacific J. Math.36 (1971), 659–669.

    MATH  MathSciNet  Google Scholar 

  11. K. H. Hofmann and P. S. MostertElements of compact Semigroups, Charles E. Merrill, Columbus, Ohio, 1966.

    MATH  Google Scholar 

  12. A. S. Householder,The Theory of Matrices in Numerical Analysis, Blaisdell, New York, 1964.

    MATH  Google Scholar 

  13. L. Mirsky,On a convex set of matrices, Arch. Math.10, (1959), 88–92.

    Article  MATH  MathSciNet  Google Scholar 

  14. L. Mirsky,Results and problems in the theory of doubly stochastic matrices, Z. Wahrscheinlichkeitstheorie und View. Gebiete1 (1963), 319–334.

    Article  MATH  MathSciNet  Google Scholar 

  15. R. Penrose,A generalized inverse for matrices, Proc. Camb. Philos. Soc.51 (1955), 406–413.

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Schreiber,On a result of S. Sherman concerning doubly stochastic matrices, Proc. Amer. Math. Soc.9 (1958), 350–353.

    Article  MATH  MathSciNet  Google Scholar 

  17. S. Schwarz,A note on the structure of the semigroup of doubly stochastic matrices, Mat. Casopis Sloven. Akad. Vied.17 (1967), 308–316.

    MathSciNet  MATH  Google Scholar 

  18. S. Sherman,A correction to “On a conjecture concerning doubly stochastic matrices”, Proc. Amer. Math. Soc.5 (1954), 988–999.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Tennessee, Knoxville, Tennessee, USA

    J. S. Montague & R. J. Plemmons

Authors
  1. J. S. Montague
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. J. Plemmons
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

His research was supported by the N. S. F. Grant GP-15943.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Montague, J.S., Plemmons, R.J. Doubly stochastic matrix equations. Israel J. Math. 15, 216–229 (1973). https://doi.org/10.1007/BF02787568

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02787568

Keywords

Search

Navigation