Geodesic
Structured condition numbers and backward errors in scalar product spaces
The electronic journal of linear algebra, Tome 15 (2006), pp. 159-177.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: The effect of structure-preserving perturbations on the solution to a linear system, matrix inversion, and distance to singularity is investigated. Particular attention is paid to linear and nonlinear structures that form Lie algebras, Jordan algebras and automorphism groups of a scalar product. These include complex symmetric, pseudo-symmetric, persymmetric, skewsymmetric, Hamiltonian, unitary, complex orthogonal and symplectic matrices. Under reasonable assumptions on the scalar product, it is shown that there is little or no difference between structured and unstructured condition numbers and distance to singularity for matrices in Lie and Jordan algebras. Hence, for these classes of matrices, the usual unstructured perturbation analysis is sufficient. It is shown that this is not true in general for structures in automorphism groups. Bounds and computable expressions for the structured condition numbers for a linear system and matrix inversion are derived for these nonlinear structures.
Classification : 15A12, 65F35, 65F15
Keywords: structured matrices, normwise structured perturbations, structured linear systems, condition number, backward error, distance to singularity, Lie algebra, Jordan algebra, automorphism group, scalar product, bilinear form, sesquilinear form, orthosymmetric 
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     title = {Structured condition numbers and backward errors in scalar product spaces},
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Tisseur, Françoise; Graillat, Stef. Structured condition numbers and backward errors in scalar product spaces. The electronic journal of linear algebra, Tome 15 (2006), pp. 159-177. http://geodesic.mathdoc.fr/item/ELA_2006__15__a16/