Form Absolute Value

  • James LottesEmail author
Part of the Springer Theses book series (Springer Theses)


This chapter presents and investigates a new absolute value of the operator associated with a sesquilinear form. Properties of the inner product and norm associated with the new absolute value generalize useful properties associated with the “energy” inner product and norm (to which they reduce in the symmetric case), without degrading for highly nonsymmetric operators. The “energy” inner product plays a key role in the analysis of symmetric problems, and its absence for nonsymmetric problems has been a significant barrier in their analysis. The new absolute value presented here fills this gap, and is a key tool used in the convergence theory of the next chapter. Existence and uniqueness are proved for operators with positive real part, while necessary and sufficient conditions are given for arbitrary matrices. Practical computation methods are also discussed.


Hilbert Space Energy Norm Polar Decomposition Positive Real Part Matrix Case 
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  1. 1.
    Bolshakov, Y., van der Mee, C.V.M., Ran, A.M., Reichstein, B., Rodman, L.: Polar decompositions in finite dimensional indefinite scalar product spaces: General theory. Linear Algebra Appl. 261, 91–141 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Donoghue, W.F.: The interpolation of quadratic norms. Acta Math. 118(1), 251–270 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Higham, N., Mehl, C., Tisseur, F.: The canonical generalized polar decomposition. SIAM J. Matrix Anal. Appl. 31(4), 2163–2180 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Higham, N.J.: Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2008)Google Scholar
  5. 5.
    Horn, R.A., Sergeichuk, V.V.: Canonical forms for complex matrix congruence and *congruence. Linear Algebra Appl. 416(2–3), 1010–1032 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Johnson, C.R., Furtado, S.: A generalization of Sylvester’s law of inertia. Linear Algebra Appl. 338, 287–290 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kato, T.: Perturbation theory for linear operators. Springer, Heidelberg (1966)CrossRefzbMATHGoogle Scholar
  8. 8.
    Kubo, F., Ando, T.: Means of positive linear operators. Math. Ann. 246(3), 205–224 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Lawson, J.D., Lim, Y.: The geometric mean, matrices, metrics, and more. Am. Math. Mon. 108(9), 797–812 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    McCarthy, J.E.: Geometric interpolation between Hilbert spaces. Ark. Mat. 30, 321–330 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Pusz, W., Woronowicz, S.L.: Functional calculus for sesquilinear forms and the purification map. Rep. Math. Phys. 8(2), 159–170 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Roşca, I.: On the Babuška Lax Milgram theorem. An. Univ. Bucureşti 38(3), 61–65 (1989)MathSciNetzbMATHGoogle Scholar

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© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Google Inc.Mountain ViewUSA

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