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Matrix Partial Orders Associated with Space Preorder

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Book cover Combinatorial Matrix Theory and Generalized Inverses of Matrices

Abstract

In this expository article, we discuss some fundamentals of well-known matrix partial orders that are closely associated with space preorder on rectangular matrices. Particularly, we consider partial order defined by space decomposition, star ordering, and minus partial order for our discussion. These relations are closely associated with comparison of column spaces and row spaces of matrices. Results associated with selected matrix relation that are known in the literature along with some interesting observations are put together. At many places, though the proofs of several results are known in the past literature, by part or completely, for better reading purpose, independent proofs are provided.

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References

  1. Anderson, W.N.: Shorted operators. SIAM J. Appl. Math. 20, 520–525 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  2. Anderson, W.N., Trapp, G.E.: Shorted operators II. SIAM J. Appl. Math. 28, 60–71 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baksalary, J.K., Mitra, S.K.: Left-star and right-star partial ordering. Linear Algebra Appl. 149, 73–89 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  4. Baksalary, J.K., Baksalary, O.M., Liu, X.: Further properties of the star, left-star, right-star and minus partial ordering. Linear Algebra Appl. 375, 83–94 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  5. Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications. Wiley-Interscience, New York (1974)

    MATH  Google Scholar 

  6. Drazin, M.P.: Natural Structures on semigroups with involution. Bull. Am. Math. Soc. 84, 139–141 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hartwig, R.E.: How to order regular elements? SIAM J. Appl. Math. 25, 1–13 (1980)

    MathSciNet  MATH  Google Scholar 

  8. Hartwig, R.E., Drazin, M.P.: Lattice properties of the ∗-order for complex matrices. J. Math. Anal. Appl. 86, 359–378 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  9. Jain, S.K., Manjunatha Prasad, K.: Right–left symmetry of aR⊕bR=(a+b)R in regular rings. J. Pure Appl. Algebra 133, 141–142 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Löewner, K.: Über monotone matrixfunktionen. Math. Z. 38, 177–216 (1934)

    Article  MathSciNet  Google Scholar 

  11. Mitra, S.K.: Fixed rank solutions of linear matrix equations. Sankhya, Ser. A 34, 387–392 (1972)

    MathSciNet  MATH  Google Scholar 

  12. Mitra, S.K.: Properties of the fundamental bordered matrix used in linear estimation. In: Kallianpur, G., et al. (eds.) Statistics and Probability: Essays in Honor of C.R. Rao, North-Holland, New York (1982)

    Google Scholar 

  13. Mitra, S.K.: The minus partial order and shorted matrix. Linear Algebra Appl. 81, 207–236 (1986)

    Article  MathSciNet  Google Scholar 

  14. Mitra, S.K.: Separation theorems. Linear Algebra Appl. 208/209, 238–256 (1994)

    Article  Google Scholar 

  15. Mitra, S.K., Bhimasankarm, P., Malik, S.B.: Matrix Partial Orders. Shorted Operators and Applications. World Scientific, Singapore (2010)

    MATH  Google Scholar 

  16. Mitra, S.K., Hartwig, R.E.: Partial orders based on outer inverses. Linear Algebra Appl. 176, 3–20 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  17. Mitra, S.K., Manjunatha Prasad, K.: The nonunique shorted matrix. Linear Algebra Appl. 237/238, 41–70 (1996)

    Article  MathSciNet  Google Scholar 

  18. Mitra, S.K., Puntanen, S.: The shorted operator statistically interpreted. Calcutta Stat. Assoc. Bull. 40, 157–160 (1990–91). Prof. H.K. Nandi memorial volume

    MathSciNet  Google Scholar 

  19. Mitra, S.K., Puntanen, S., Styan, G.P.H.: Shorted matrices and their applications in linear statistical models, a review. In: Tiit, M., et al. (eds.) New Trends in Probability and Statistics. Proceedings of the 5th Tartu Conference on Multivariate Statistics, pp. 289–311. TEV, Vilnius (1995)

    Google Scholar 

  20. Mitra, S.K., Puri, M.L.: Shorted operators and generalized inverses of matrices. Linear Algebra Appl. 25, 45–56 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  21. Mitra, S.K., Puri, M.L.: Shorted matrix: An extended concept and some applications. Linear Algebra Appl. 42, 57–79 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  22. Moore, E.H.: On the reciprocal of the general algebraic matrix. Bull. Am. Math. Soc. 26, 394–395 (1920)

    Google Scholar 

  23. Nambooripad, K.S.S.: The natural partial order on a regular semigroup. Proc. Edinb. Math. Soc. 23, 249–260 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. Penrose, R.: Generalized inverse for matrices. Proc. Camb. Philos. Soc. 51, 406–413 (1955)

    Article  MathSciNet  MATH  Google Scholar 

  25. Rao, C.R.: A note on a generalized inverse of a matrix with applications to problems in mathematical statistics. J. R. Stat. Soc. B 24, 152–158 (1962)

    MATH  Google Scholar 

  26. Rao, C.R.: Unified theory of linear estimation. Sankhya, Ser. A 33, 371–394 (1971)

    MathSciNet  MATH  Google Scholar 

  27. Rao, C.R., Mitra, S.K.: Generalized Inverse of Matrices and Its Applications. Wiley, New York (1971)

    MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Statistics, Manipal University, Manipal, 576104, India

    K. Manjunatha Prasad, K. S. Mohana & Y. Santhi Sheela

  2. Department of Mathematics, Manipal Institute of Technology, Manipal, 576104, India

    K. S. Mohana

Authors
  1. K. Manjunatha Prasad
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  2. K. S. Mohana
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  3. Y. Santhi Sheela
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Corresponding author

Correspondence to K. S. Mohana .

Editor information

Editors and Affiliations

  1. Indian Statistical Institute, S.J.S. Sansanwal Marg 7, New Delhi, 110016, India

    Ravindra B. Bapat

  2. Hamilton Institute, National University of Ireland, Kilkenny, Maynooth, Co. Kildar, Kilkenny, Ireland

    Steve J. Kirkland

  3. Department of Statistics, Manipal University, manipal.edu, Manipal, 576104, Karnataka, India

    K. Manjunatha Prasad

  4. , School of Information Sciences, University of Tampere, Kalevantie 4,, Tampere, 33014, Finland

    Simo Puntanen

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Manjunatha Prasad, K., Mohana, K.S., Santhi Sheela, Y. (2013). Matrix Partial Orders Associated with Space Preorder. In: Bapat, R., Kirkland, S., Prasad, K., Puntanen, S. (eds) Combinatorial Matrix Theory and Generalized Inverses of Matrices. Springer, India. https://doi.org/10.1007/978-81-322-1053-5_17

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