Structured conditioning of matrix functions

Abstract

The existing theory of conditioning for matrix functions f(X): C^nxn→ C^nxn does not cater for structure in the matrix X. An extension of this theory is presented in which when X has structure, all perturbations of X are required to have the same structure. Two classes of structure matrices are considered, those comprising the Jordan algebra J and the Lie algebra L associated with a nondegenerate bilinear or sesquilinear form on Rⁿ or Cⁿ. Example of such classes are the summetric, skew-symmetric, Hamiltonian and skew-Hamiltonian matrices. Structured condition numbers are defined for these two classes. Under certain conditions on the underlying scalar product, explicit representations are given for the structured condition numbers. Comparisions between the unstructured and structured condition numbers are then made. When the underlying scalar product is a sesquilinear form, it is shown that there is no difference between the values of the two condition numbers for (i) all functions of X ∈ J, and (ii) odd and even fuctions of X ∈ L. When the underlying scalar prodcut is a bilinear form then equality is not guaranteed in all these cases. When equality is not guaranteed, bounds are obtained for the ration of the unstructured and structured condition numbers.