A new efficient approach to the characterization of D-stable matrices

The concept of D-stability is relevant for stable square matrices of any order, especially when they appear in ordinary differential systems modeling physical problems. Indeed, D-stability was treated from different points of view in the last 50 years, but the problem of characterization of a general D-stable matrix was solved for low-order matrices only (ie, up to order 4). Here, a new approach is proposed within the context of numerical linear algebra. Starting from a known necessary and sufficient condition, other simpler equivalent necessary and sufficient conditions for D-stability are proved. Such conditions turn out to be computationally more appealing for symbolic software, as discussed in the reported examples. Therefore, a new symbolic method is proposed to characterize matrices of order greater than 4, and then it is used in some numerical examples, given in details.