A Note on Families of Stable Matrices

Many topological properties of the whole family of stable polynomials and matrices have been studied in a number of publications (see e.g., [1,3–7,11,12]). The present paper continues these investigations with the aim of strengthening these results in the sense that instead of simple-connectivity we show stronger property of contractibility of the various families of stable matrices from this point of view. We now recall some topological notions needed below. A metric (or topological) space which can be reduced to one of its points by a continuous deformation is said to be contractible. In other words let X be a metric space, Y ⊂ X and x0 ∈ Y . Then Y is called contractible to the point x0 ∈ Y , if there exists a continuous function F : Y × [0, 1] → Y such that