Componentwise Distance to Singularity

A perturbation matrix A = A is considered, where A 2 IR and 0 2 IR. The matrix A is singular i A contains a real singular matrix. A problem is to decide if A is singular or nonsingular, a NP-hard problem. The decision can be made by the computation of the componentwise distance to the nearest singular matrix de ned on the basis of the real spectral radius, and by the solution of 4 eigenvalue problems. Theorem 6 gives a new computation basis, a natural way to the \componentwise distance ..." de nition, and a motivation to rename this in radius of singularity denoted by sir(A; ). This new way shows: (i) sir results from a real nonnegative eigensolution of a nonlinear mapping, (ii) sir has a norm representation, (iii) sir can be computed by 2 1 nonnegative eigensolutions of the nonlinear mapping, (iv) for the special case = pq ; 0 p; q 2 IR a formula for a computation of sir is given, also a trivial algorithm for the computation, and some examples as demonstration.