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

  • Yonghui Liu
  • Yongge Tian


We introduce a simultaneous decomposition for a matrix triplet (A,B,C ), where AA 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 ABXC±(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 ABXC−(BXC) with respect to X. As applications, we derive the extremal ranks and inertias of the matrix expression DCXC subject to Hermitian solutions of a consistent matrix equation AXA =B, as well as the extremal ranks and inertias of the Hermitian Schur complement DB 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.


Hermitian matrix Rank Inertia Generalized inverse Schur complement Inequality 


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  1. 1.
    Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2003) MATHGoogle Scholar
  2. 2.
    Bernstein, D.S.: Matrix Mathematics: Theory, Facts and Formulas, 2nd edn. Princeton University Press, Princeton (2009) MATHGoogle Scholar
  3. 3.
    Hogben, L.: Handbook of Linear Algebra. Chapman & Hall/CRC, New York (2007) MATHGoogle Scholar
  4. 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) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Tian, Y., Liu, Y.: Extremal ranks of some symmetric matrix expressions with applications. SIAM J. Matrix Anal. Appl. 28, 890–905 (2006) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Liu, Y., Tian, Y.: More on extremal ranks of the matrix expressions ABX±X B with statistical applications. Numer. Linear Algebra Appl. 15, 307–325 (2008) MATHCrossRefMathSciNetGoogle Scholar
  7. 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) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Liu, Y., Tian, Y.: A simultaneous decomposition of a matrix triplet with applications. Numer. Linear Algebra Appl. doi: 10.1002/nla.701
  9. 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) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Tian, Y.: Completing block matrices with maximal and minimal ranks. Linear Algebra Appl. 321, 327–345 (2000) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Tian, Y.: Rank equalities related to outer inverses of matrices and applications. Linear Multilinear Algebra 49, 269–288 (2002) CrossRefGoogle Scholar
  12. 12.
    Tian, Y.: The minimum rank of a 3×3 partial block matrix. Linear Multilinear Algebra 50, 125–131 (2002) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Tian, Y.: Upper and lower bounds for ranks of matrix expressions using generalized inverses. Linear Algebra Appl. 355, 187–214 (2002) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Tian, Y.: The maximal and minimal ranks of some expressions of generalized inverses of matrices. Southeast Asian Bull. Math. 25, 745–755 (2002) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Tian, Y.: Ranks of solutions of the matrix equation AXB=C. Linear Multilinear Algebra 51, 111–125 (2003) CrossRefMathSciNetGoogle Scholar
  16. 16.
    Tian, Y.: More on maximal and minimal ranks of Schur complements with applications. Appl. Math. Comput. 152, 175–192 (2004) CrossRefGoogle Scholar
  17. 17.
    Tian, Y.: Rank equalities for block matrices and their Moore–Penrose inverses. Houston J. Math. 30, 483–510 (2004) MATHMathSciNetGoogle Scholar
  18. 18.
    Tian, Y.: Equalities and inequalities for inertias of Hermitian matrices with applications. Linear Algebra Appl. 433, 263–296 (2010) MATHCrossRefMathSciNetGoogle Scholar
  19. 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) MathSciNetGoogle Scholar
  20. 20.
    Tian, Y.: Extremal ranks of a quadratic matrix expression with applications. Linear Multilinear Algebra (2010, in press) Google Scholar
  21. 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. 22.
    Tian, Y., Cheng, S.: The maximal and minimal ranks of ABXC with applications. New York J. Math. 9, 345–362 (2003) MATHMathSciNetGoogle Scholar
  23. 23.
    Tian, Y., Styan, G.P.H.: Rank equalities for idempotent and involutory matrices. Linear Algebra Appl. 335, 101–117 (2001) MATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Tian, Y., Styan, G.P.H.: Rank equalities for idempotent matrices with applications. J. Comput. Appl. Math. 191, 77–97 (2006) MATHCrossRefMathSciNetGoogle Scholar
  25. 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) MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Paige, C.C., Saunders, M.A.: Towards a generalized singular value decomposition. SIAM J. Numer. Anal. 18, 398–405 (1981) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Marsaglia, G., Styan, G.P.H.: Equalities and inequalities for ranks of matrices. Linear Multilinear Algebra 2, 269–292 (1974) CrossRefMathSciNetGoogle Scholar
  28. 28.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press, Cambridge (1985) MATHGoogle Scholar
  29. 29.
    Mirsky, L.: An Introduction to Linear Algebra. Dover, New York (1990). Second Corrected Reprint Edition MATHGoogle Scholar
  30. 30.
    Haynsworth, E.V.: Determination of the inertia of a partitioned Hermitian matrix. Linear Algebra Appl. 1, 73–81 (1968) MATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    Haynsworth, E.V., Ostrowski, A.M.: On the inertia of some classes of partitioned matrices. Linear Algebra Appl. 1, 299–316 (1968) MATHCrossRefMathSciNetGoogle Scholar
  32. 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) CrossRefMathSciNetGoogle Scholar
  33. 33.
    Yasuda, K., Skelton, R.E.: Assigning controllability, and observability Gramians in feedback control. J. Guid. Control Dyn. 14, 878–885 (1990) CrossRefMathSciNetGoogle Scholar
  34. 34.
    Gahinet, P., Apkarian, P.: A linear matrix inequality approach to H control. Internat. J. Robust Nonlinear Control 4, 421–448 (1994) MATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Iwasaki, T., Skelton, R.E.: All controllers for the general Open image in new window control problem: LMI existence conditions and state space formulas. Automatica 30, 1307–1317 (1994) MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    Scherer, C.W.: A complete algebraic solvability test for the nonstrict Lyapunov inequality. Syst. Control Lett. 25, 327–335 (1995) MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Skelton, R.E., Iwasaki, T., Grigoriadis, K.M.: A Unified Algebraic Approach to Linear Control Design. Taylor & Francis, London (1998) Google Scholar
  38. 38.
    Groß, J.: A note on the general Hermitian solution to AXA =B. Bull. Malays. Math. Soc. (2nd Ser.) 21, 57–62 (1998) MATHGoogle Scholar
  39. 39.
    Cain, B.E.: The inertia of a Hermitian matrix having prescribed diagonal blocks. Linear Algebra Appl. 37, 173–180 (1981) MATHCrossRefMathSciNetGoogle Scholar
  40. 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) MATHCrossRefMathSciNetGoogle Scholar
  41. 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) MATHCrossRefGoogle Scholar
  42. 42.
    Cohen, N., Dancis, J.: Inertias of block band matrix completions. SIAM J. Matrix Anal. Appl. 19, 583–612 (1998) MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Cohen, N., Johnson, C.R., Rodman, L., Woerdeman, H.J.: Rank completions of partial matrices. Oper. Theory Adv. Appl. 40, 165–185 (1989) MathSciNetGoogle Scholar
  44. 44.
    Constantinescu, T., Gheondea, A.: The negative signature of some Hermitian matrices. Linear Algebra Appl. 178, 17–42 (1993) MATHCrossRefMathSciNetGoogle Scholar
  45. 45.
    Da Fonseca, C.M.: The inertia of certain Hermitian block matrices. Linear Algebra Appl. 274, 193–210 (1998) MATHCrossRefMathSciNetGoogle Scholar
  46. 46.
    Dancis, J.: The possible inertias for a Hermitian matrix and its principle submatrices. Linear Algebra Appl. 85, 121–151 (1987) MATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    Dancis, J.: Poincaré’s inequalities and Hermitian completions of certain partial matrices. Linear Algebra Appl. 167, 219–225 (1992) CrossRefMathSciNetGoogle Scholar
  48. 48.
    Geelen, J.F.: Maximum rank matrix completion. Linear Algebra Appl. 288, 211–217 (1999) MATHCrossRefMathSciNetGoogle Scholar
  49. 49.
    Gheondea, A.: One-step completions of Hermitian partial matrices with minimal negative signature. Linear Algebra Appl. 173, 99–114 (1992) MATHCrossRefMathSciNetGoogle Scholar
  50. 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) MATHGoogle Scholar
  51. 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) MATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    Maddocks, J.H.: Restricted quadratic forms, inertia theorems and the Schur complement. Linear Algebra Appl. 108, 1–36 (1988) MATHCrossRefMathSciNetGoogle Scholar
  53. 53.
    Takahashi, K.: Invertible completions of operator matrices. Integral Equ. Oper. Theory 21, 355–361 (1995) MATHCrossRefGoogle Scholar
  54. 54.
    Woerdeman, H.J.: Minimal rank completions for block matrices. Linear Algebra Appl. 121, 105–122 (1989) MATHCrossRefMathSciNetGoogle Scholar
  55. 55.
    Woerdeman, H.J.: Toeplitz minimal rank completions. Linear Algebra Appl. 202, 267–278 (1994) MATHCrossRefMathSciNetGoogle Scholar
  56. 56.
    Woerdeman, H.J.: Minimal rank completions of partial banded matrices. Linear Multilinear Algebra 36, 59–68 (1993) MATHCrossRefMathSciNetGoogle Scholar
  57. 57.
    Woerdeman, H.J.: Hermitian and normal completions. Linear Multilinear Algebra 42, 239–280 (1997) MATHCrossRefMathSciNetGoogle Scholar
  58. 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) CrossRefGoogle Scholar
  59. 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. 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. 61.
    Baksalary, J.K., Kala, R.: Symmetrizers of matrices. Linear Algebra Appl. 35, 51–62 (1981) MATHCrossRefMathSciNetGoogle Scholar
  62. 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) MATHCrossRefMathSciNetGoogle Scholar
  63. 63.
    Khatskevich, V.A., Ostrovskii, M.I., Shulman, V.S.: Quadratic inequalities for Hilbert space operators. Integral. Equ. Oper. Theory 59, 19–34 (2007) MATHCrossRefMathSciNetGoogle Scholar

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© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Applied MathematicsShanghai Finance UniversityShanghaiChina
  2. 2.China Economics and Management AcademyCentral University of Finance and EconomicsBeijingChina

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