Derivatives of Multilinear Functions of Matrices

  • Priyanka Grover


Perturbation or error bounds of functions have been of great interest for a long time. If the functions are differentiable, then the mean value theorem and Taylor’s theorem come handy for this purpose.



This article is based on my talk at Indo-French Seminar on Matrix Information Geometries, funded by Indo-French Centre for the Promotion of Advanced Research. I am thankful to my supervisor Prof. Rajendra Bhatia and other participants of the Seminar for their useful comments and suggestions.


  1. 1.
    Bapat, R.B.: Inequalities for mixed Schur functions. Linear Algebra Appl. 83, 143–149 (1986)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Bapat, R.B.: Mixed discriminants of positive semidefinite matrices. Linear Algebra Appl. 126, 107–124 (1989)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bhatia, R.: Matrix Analysis. Springer, New York (1997)CrossRefGoogle Scholar
  4. 4.
    Bhatia, R.: Positive Definite Matrices. Princeton University Press, New Jersey (2007)Google Scholar
  5. 5.
    Bhatia, R.: Perturbation Bounds for Matrix Eigenvalues, SIAM, Philadelphia (2007). Expanded reprint of 1987 editionGoogle Scholar
  6. 6.
    Bhatia, R.: Variation of symmetric tensor powers and permanents. Linear Algebra Appl. 62, 269–276 (1984)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Bhatia, R., Dias da Silva, J.A.: Variation of induced linear operators. Linear Algebra Appl. 341, 391–402 (2002)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bhatia, R., Friedland, S.: Variation of Grassman powers and spectra. Linear Algebra Appl. 40, 1–18 (1981)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Bhatia, R., Jain, T.: Higher order derivatives and perturbation bounds for determinants. Linear Algebra Appl. 431, 2102–2108 (2009)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Dieudonn\(\acute{\text{ e}}\) J.: Foundations of Modern Analysis. Academic Press, New York (1960)Google Scholar
  11. 11.
    Grover P.: Derivatives and perturbation bounds for symmetric tensor powers of matrices, arXiv:1102.2414v2 [math.FA]Google Scholar
  12. 12.
    Gurvits, L.: The Van der Waerden conjecture for mixed discriminants. Adv. Math. 200, 435–454 (2006)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1991)MATHCrossRefGoogle Scholar
  14. 14.
    Jain, T.: Derivatives for antisymmetric tensor powers and perturbation bounds. Linear Algebra Appl. 435, 1111–1121 (2011)MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Marcus, M., Minc, H.: A Survey of Matrix Theory and Matrix Inequalities. Dover Publications, New York (1992). Reprint of 1964 editionGoogle Scholar
  16. 16.
    Merris, R.: Multilinear Algebra. Gordon and Breach Science Publishers, Singapore (1997)MATHGoogle Scholar
  17. 17.
    Minc, H.: Permanents. Addison-Wesley Publishing Company, Massachusetts (1970)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Theoretical Statistics and Mathematics UnitIndian Statistical Institute Delhi CentreNew DelhiIndia

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