Skip to main content
SpringerLink
Log in

Max-Min Problems on the Ranks and Inertias of the Matrix Expressions A−BXC±(BXC)∗ with Applications

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

We introduce a simultaneous decomposition for a matrix triplet (A,B,C ∗), where A=±A ∗ and (⋅)∗ denotes the conjugate transpose of a matrix, and use the simultaneous decomposition to solve some conjectures on the maximal and minimal values of the ranks of the matrix expressions A−BXC±(BXC)∗ with respect to a variable matrix X. In addition, we give some explicit formulas for the maximal and minimal values of the inertia of the matrix expression A−BXC−(BXC)∗ with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression D−CXC ∗ subject to Hermitian solutions of a consistent matrix equation AXA ∗=B, as well as the extremal ranks and inertias of the Hermitian Schur complement D−B ∗ A ∼ B with respect to a Hermitian generalized inverse A ∼ of A. Various consequences of these extremal ranks and inertias are also presented in the paper.

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.

Similar content being viewed by others

References

  1. Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003)

    MATH  Google Scholar 

  2. Bernstein, D.S.: Matrix Mathematics: Theory, Facts and Formulas, 2nd edn. Princeton University Press, Princeton (2009)

    MATH  Google Scholar 

  3. Hogben, L.: Handbook of Linear Algebra. Chapman & Hall/CRC, New York (2007)

    MATH  Google Scholar 

  4. Barrett, W., Hall, H.T., Loewy, R.: The inverse inertia problem for graphs: cut vertices, trees, and a counterexample. Linear Algebra Appl. 431, 1147–1191 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Tian, Y., Liu, Y.: Extremal ranks of some symmetric matrix expressions with applications. SIAM J. Matrix Anal. Appl. 28, 890–905 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  6. Liu, Y., Tian, Y.: More on extremal ranks of the matrix expressions A−BX±X ∗ B ∗ with statistical applications. Numer. Linear Algebra Appl. 15, 307–325 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  7. Liu, Y., Tian, Y.: Extremal ranks of submatrices in an Hermitian solution to the matrix equation AXA ∗=B with applications. J. Appl. Math. Comput. 32, 289–301 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  8. Liu, Y., Tian, Y.: A simultaneous decomposition of a matrix triplet with applications. Numer. Linear Algebra Appl. doi:10.1002/nla.701

  9. Liu, Y., Tian, Y., Takane, Y.: Ranks of Hermitian and skew-Hermitian solutions to the matrix equation AXA ∗=B. Linear Algebra Appl. 431, 2359–2372 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  10. Tian, Y.: Completing block matrices with maximal and minimal ranks. Linear Algebra Appl. 321, 327–345 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  11. Tian, Y.: Rank equalities related to outer inverses of matrices and applications. Linear Multilinear Algebra 49, 269–288 (2002)

    Article  Google Scholar 

  12. Tian, Y.: The minimum rank of a 3×3 partial block matrix. Linear Multilinear Algebra 50, 125–131 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  13. Tian, Y.: Upper and lower bounds for ranks of matrix expressions using generalized inverses. Linear Algebra Appl. 355, 187–214 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  14. Tian, Y.: The maximal and minimal ranks of some expressions of generalized inverses of matrices. Southeast Asian Bull. Math. 25, 745–755 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  15. Tian, Y.: Ranks of solutions of the matrix equation AXB=C. Linear Multilinear Algebra 51, 111–125 (2003)

    Article  MathSciNet  Google Scholar 

  16. Tian, Y.: More on maximal and minimal ranks of Schur complements with applications. Appl. Math. Comput. 152, 175–192 (2004)

    Article  Google Scholar 

  17. Tian, Y.: Rank equalities for block matrices and their Moore–Penrose inverses. Houston J. Math. 30, 483–510 (2004)

    MATH  MathSciNet  Google Scholar 

  18. Tian, Y.: Equalities and inequalities for inertias of Hermitian matrices with applications. Linear Algebra Appl. 433, 263–296 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  19. Tian, Y.: Rank and inertia of submatrices of the Moore–Penrose inverse of a Hermitian matrix. Electron. J. Linear Algebra 20, 226–240 (2010)

    MathSciNet  Google Scholar 

  20. Tian, Y.: Extremal ranks of a quadratic matrix expression with applications. Linear Multilinear Algebra (2010, in press)

  21. Tian, Y.: Completing a block Hermitian matrix with maximal and minimal ranks and inertias. Electron. Linear Algebra Appl. 21, 124–141 (2010)

    Google Scholar 

  22. Tian, Y., Cheng, S.: The maximal and minimal ranks of A−BXC with applications. New York J. Math. 9, 345–362 (2003)

    MATH  MathSciNet  Google Scholar 

  23. Tian, Y., Styan, G.P.H.: Rank equalities for idempotent and involutory matrices. Linear Algebra Appl. 335, 101–117 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  24. Tian, Y., Styan, G.P.H.: Rank equalities for idempotent matrices with applications. J. Comput. Appl. Math. 191, 77–97 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  25. Tian, Y., Takane, Y.: The inverse of any two-by-two nonsingular partitioned matrix and three matrix inverse completion problems. Comput. Math. Appl. 57, 1294–1304 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  26. Paige, C.C., Saunders, M.A.: Towards a generalized singular value decomposition. SIAM J. Numer. Anal. 18, 398–405 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  27. Marsaglia, G., Styan, G.P.H.: Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2, 269–292 (1974)

    Article  MathSciNet  Google Scholar 

  28. Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985)

    MATH  Google Scholar 

  29. Mirsky, L.: An Introduction to Linear Algebra. Dover, New York (1990). Second Corrected Reprint Edition

    MATH  Google Scholar 

  30. Haynsworth, E.V.: Determination of the inertia of a partitioned Hermitian matrix. Linear Algebra Appl. 1, 73–81 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  31. Haynsworth, E.V., Ostrowski, A.M.: On the inertia of some classes of partitioned matrices. Linear Algebra Appl. 1, 299–316 (1968)

    Article  MATH  MathSciNet  Google Scholar 

  32. Fujioka, H., Hara, S.: State covariance assignment problem with measurement noise a unified approach based on a symmetric matrix equation. Linear Algebra Appl. 203/204, 579–605 (1994)

    Article  MathSciNet  Google Scholar 

  33. Yasuda, K., Skelton, R.E.: Assigning controllability, and observability Gramians in feedback control. J. Guid. Control Dyn. 14, 878–885 (1990)

    Article  MathSciNet  Google Scholar 

  34. Gahinet, P., Apkarian, P.: A linear matrix inequality approach to H ∞ control. Internat. J. Robust Nonlinear Control 4, 421–448 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  35. Iwasaki, T., Skelton, R.E.: All controllers for the general control problem: LMI existence conditions and state space formulas. Automatica 30, 1307–1317 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  36. Scherer, C.W.: A complete algebraic solvability test for the nonstrict Lyapunov inequality. Syst. Control Lett. 25, 327–335 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  37. Skelton, R.E., Iwasaki, T., Grigoriadis, K.M.: A Unified Algebraic Approach to Linear Control Design. Taylor & Francis, London (1998)

    Google Scholar 

  38. Groß, J.: A note on the general Hermitian solution to AXA ∗=B. Bull. Malays. Math. Soc. (2nd Ser.) 21, 57–62 (1998)

    MATH  Google Scholar 

  39. Cain, B.E.: The inertia of a Hermitian matrix having prescribed diagonal blocks. Linear Algebra Appl. 37, 173–180 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  40. Cain, B.E., de Sá, E.M.: The inertia of a Hermitian matrix having prescribed complementary principal submatrices. Linear Algebra Appl. 37, 161–171 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  41. Cain, B.E., de Sá, E.M.: The inertia of Hermitian matrices with a prescribed 2×2 block decomposition. Linear Multilinear Algebra 31, 119–130 (1992)

    Article  MATH  Google Scholar 

  42. Cohen, N., Dancis, J.: Inertias of block band matrix completions. SIAM J. Matrix Anal. Appl. 19, 583–612 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  43. Cohen, N., Johnson, C.R., Rodman, L., Woerdeman, H.J.: Rank completions of partial matrices. Oper. Theory Adv. Appl. 40, 165–185 (1989)

    MathSciNet  Google Scholar 

  44. Constantinescu, T., Gheondea, A.: The negative signature of some Hermitian matrices. Linear Algebra Appl. 178, 17–42 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  45. Da Fonseca, C.M.: The inertia of certain Hermitian block matrices. Linear Algebra Appl. 274, 193–210 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  46. Dancis, J.: The possible inertias for a Hermitian matrix and its principle submatrices. Linear Algebra Appl. 85, 121–151 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  47. Dancis, J.: Poincaré’s inequalities and Hermitian completions of certain partial matrices. Linear Algebra Appl. 167, 219–225 (1992)

    Article  MathSciNet  Google Scholar 

  48. Geelen, J.F.: Maximum rank matrix completion. Linear Algebra Appl. 288, 211–217 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  49. Gheondea, A.: One-step completions of Hermitian partial matrices with minimal negative signature. Linear Algebra Appl. 173, 99–114 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  50. Gohberg, I., Kaashoek, M.A., van Schagen, F.: Partially Specified Matrices and Operators: Classification, Completion, Applications. Operator Theory Adv. Appl., vol. 79. Birkhauser, Boston (1995)

    MATH  Google Scholar 

  51. Grone, J., Johnson, C.R., de Sá, E.M., Wolkowitz, H.: Positive definite completions of partial Hermitian matrices. Linear Algebra Appl. 58, 109–124 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  52. Maddocks, J.H.: Restricted quadratic forms, inertia theorems and the Schur complement. Linear Algebra Appl. 108, 1–36 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  53. Takahashi, K.: Invertible completions of operator matrices. Integral Equ. Oper. Theory 21, 355–361 (1995)

    Article  MATH  Google Scholar 

  54. Woerdeman, H.J.: Minimal rank completions for block matrices. Linear Algebra Appl. 121, 105–122 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  55. Woerdeman, H.J.: Toeplitz minimal rank completions. Linear Algebra Appl. 202, 267–278 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  56. Woerdeman, H.J.: Minimal rank completions of partial banded matrices. Linear Multilinear Algebra 36, 59–68 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  57. Woerdeman, H.J.: Hermitian and normal completions. Linear Multilinear Algebra 42, 239–280 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  58. Harvey, N.J.A., Karger, D.R., Yekhanin, S.: The complexity of matrix completion. In: Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithm, Association for Computing Machinery, pp. 1103–1111. SIAM, New York (2006)

    Chapter  Google Scholar 

  59. Laurent, M.: Matrix completion problems. In: Floudas, C., Pardalos, P. (eds.) The Encyclopedia of Optimization, vol. III, pp. 221–229. Kluwer Academic, Dordrecht (2001)

    Google Scholar 

  60. Mahajan, M., Sarma, J.: On the complexity of matrix rank and rigidity. In: Lecture Notes in Computer Science, vol. 4649, pp. 269–280. Springer, New York (2007)

    Google Scholar 

  61. Baksalary, J.K., Kala, R.: Symmetrizers of matrices. Linear Algebra Appl. 35, 51–62 (1981)

    Article  MATH  MathSciNet  Google Scholar 

  62. Venkaiah, V.Ch., Sen, S.K.: Computing a matrix symmetrizer exactly using modified multiple modulus residue arithmetic. J. Comput. Appl. Math. 21, 27–40 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  63. Khatskevich, V.A., Ostrovskii, M.I., Shulman, V.S.: Quadratic inequalities for Hilbert space operators. Integral. Equ. Oper. Theory 59, 19–34 (2007)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Shanghai Finance University, Shanghai, 201209, China

    Yonghui Liu

  2. China Economics and Management Academy, Central University of Finance and Economics, Beijing, 100081, China

    Yongge Tian

Authors
  1. Yonghui Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yongge Tian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongge Tian.

Additional information

Communicated by F. Zirilli.

We wish to thank an Associate Editor and referees for helpful comments and suggestions on this paper. The research of the first author was supported by the Shanghai Municipal Natural Science Foundation (10ZR1420600); Foundation of Shanghai Municipal Education Commission (11zz182); Shanghai Talent Development Funds (078); Leading Academic Discipline Project of Shanghai Municipal Education Commission (J51601).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Liu, Y., Tian, Y. Max-Min Problems on the Ranks and Inertias of the Matrix Expressions A−BXC±(BXC)∗ with Applications. J Optim Theory Appl 148, 593–622 (2011). https://doi.org/10.1007/s10957-010-9760-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-010-9760-8

Keywords

Search

Navigation