Form Absolute Value
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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.
KeywordsHilbert Space Energy Norm Polar Decomposition Positive Real Part Matrix Case
- 4.Higham, N.J.: Functions of Matrices: Theory and Computation. Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2008)Google Scholar