ELA, Volume 11, pp. 132-161, June 2004, abstract.

Structured Conditioning of Matrix Functions

Philip I. Davies

The existing theory of conditioning for matrix functions 
f(X) from n-by-n complex matrices into n-by-n complex matrices
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 structured matrices are considered, those 
comprising the Jordan algebra J and the Lie algebra L associated
with a nondegenerate bilinear or sesquilinear form on R^n or C^n. 
Examples of such classes are the symmetric, 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. Comparisons 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 in J, and (ii) odd and even functions of 
X in L. When the underlying scalar product is a bilinear form then 
equality is not guaranteed in all these cases. Where equality is not 
guaranteed, bounds are obtained for the  ratio of the unstructured 
and structured condition numbers.