Structured Perturbations Part II: Componentwise Distances

• Authors: Siegfried M. Rump
• Venue: SIAM Journal on Matrix Analysis and Applications
• Year: 2003

In the second part of this paper we study condition numbers with respect to componentwise perturbations in the input data for linear systems and for matrix inversion, and the distance to the nearest singular matrix. The structures under investigation are linear structures, namely symmetric, persymmetric, skewsymmetric, symmetric Toeplitz, general Toeplitz, circulant, Hankel, and persymmetric Hankel structures. We give various formulas and estimations for the condition numbers. For all structures mentioned except circulant structures we give explicit examples of linear systems $A_{\varepsilon}x=b$ with parameterized matrix $A_{\varepsilon}$ such that the unstructured componentwise condition number is ${\mathcal O}(\varepsilon^{-1})$ and the structured componentwise condition number is ${\mathcal O}(1)$. This is true for the important case of componentwise relative perturbations in the matrix and in the right-hand side. We also prove corresponding estimations for circulant structures. Moreover, bounds for the condition number of matrix inversion are given. Finally, we give for all structures mentioned above explicit examples of parameterized (structured) matrices $A_{\varepsilon}$ such that the (componentwise) condition number of matrix inversion is ${\mathcal O}(\varepsilon^{-1})$, but the componentwise distance to the nearest singular matrix is ${\mathcal O}(1)$. This is true for componentwise relative perturbations. It shows that, unlike the normwise case, there is no reciprocal proportionality between the componentwise condition number and the distance to the nearest singular matrix.