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Matrix Inequalities and Twisted Inner Products

  • Thomas H. Pate
Part of the Operator Theory: Advances and Applications book series (OT, volume 202)

Abstract

We will demonstrate that several known inequalities involving generalized Schur functions, also known as generalized matrix functions, follow from either the Cauchy-Schwartz inequality, or from certain monotonicity relations that exist between inner products on spaces of multilinear functions. Connections between our inner products and permanent inequalities are presented, and a connection to some unresolved problems in partial differential equations is indicated.

Keywords

Matrix inequalities inner products tensors multilinear functions tensor inequalities 

Mathematics Subject Classification (2000)

15A15 15A45 15A63 15A69 

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References

  1. [1]
    P. Heyfron, Immanant dominance orderings for hook partitions, Linear and Multilinear Algebra, (1) 24 (1988), 65–78.CrossRefMathSciNetGoogle Scholar
  2. [2]
    R.A. Horn and C.R. Johnson, Matrix Analysis, Cambridge University Press, 1985.Google Scholar
  3. [3]
    E.H. Lieb, Proofs of some conjectures on permanents, J. Math. and Mech., 16 (1966), 127–134.zbMATHMathSciNetGoogle Scholar
  4. [4]
    M. Marcus, Multilinear methods in linear algebra, Linear Algebra and Its Applications, 150 (1991), 41–59.CrossRefMathSciNetGoogle Scholar
  5. [5]
    J.W. Neuberger, Norm of symmetric product compared with norm of tensor product, Linear and Multilinear Algebra, 2 (1974), 115–122.CrossRefMathSciNetGoogle Scholar
  6. [6]
    J.W. Neuberger, Tensor products and successive approximations for partial differential equations, Israel Journal of Mathematics, 6 (1968), 121–132.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    J.W. Neuberger, An iterative method for solving non-linear partial differential equations, Advances in Mathematics, (2) 19 (1976), 245–265.zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    J.W. Neuberger, A resolvent for an iteration method for nonlinear partial differential equations, Transactions of the American Mathematical Society, 226 (1977), 321–343.zbMATHCrossRefMathSciNetGoogle Scholar
  9. [9]
    J.W. Neuberger, Sobolev Gradients and Differential Equations, Springer Lecture Notes in Mathematics, 1670, Springer, 1997.Google Scholar
  10. [10]
    T.H. Pate, Lower bounds for the norm of the symmetric product, Linear Algebra and Its Applications, (3) 14 (1976), 285–291.Google Scholar
  11. [11]
    T.H. Pate, Tensor products, symmetric products, and permanents of positive semidefinite Hermitian matrices, Linear and Multilinear Algebra, (1–4) 31 (1992), 27–36.zbMATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    T.H. Pate, On Permanental compounds, Linear Algebra and Its Applications, (5-6) 429 (2008), 1093–1101.CrossRefMathSciNetGoogle Scholar
  13. [13]
    T.H. Pate, Group algebras, monotonicity, and the Lieb Permanent Inequality, Linear and Multilinear Algebra, 40 (1996), 207–220.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    T.H. Pate, Some improvements in Neuberger’s iteration procedure for solving partial differential equations, Journal of Differential Equations, (1) 34 (1979), 261–272.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [15]
    T.H. Pate, An application of the Neuberger iteration procedure to the constant coefficient linear partial differential equation, Journal of Mathematical Analysis and Applications, (2) 72 (1979), 771–782.zbMATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    T.H. Pate, A new lower bound inequality for the norm of the symmetric product, Linear and Multilinear Algebra (1) 57 (2009), 87–102.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [17]
    T.H. Pate, Row appending maps, Ψ-functions, and immanant inequalities for Hermitian positive semi-definite matrices, Proceedings of the London Mathematical Society, (2) 76 (1998), 307–358.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    T.H. Pate, A generalized Neuberger identity for the inner product of symmetric products of tensors, Linear and Multilinear Algebra, (5) 56 (2008), 555–563.zbMATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    T.H. Pate, Twisted inner products and contraction inequalities on spaces of contravariant and covariant tensors, Linear Algebra and Its Applications, (7) 429 (2008), 1489–1503.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2010

Authors and Affiliations

  • Thomas H. Pate
    • 1
  1. 1.Mathematics DepartmentAuburn UniversityAuburnUSA

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