Some applications of matrices to location of zeros of polynomials
Stephen Mark Barnett · International Journal of Control · 1973
In previous papers classical theorems on location of zeros of a polynomial with respect to the left half plane Γ1 or the unit circle Γ2 have been reformulated more simply in terms of appropriate companion matrices. It is shown how this work can be extended to the problem of zero location with respect to more general regions Γ of the complex plane. The first approach is to apply the bilinear transformation to the given polynomial, so that for example Γ1 can be mapped into Γ2 and a matrix representation of this is derived. An alternative method is discussed which relies on transformation of Γ into Γ2. Some examples illustrate how any theorem involving Hurwitz-typc minors can be expressed in companion matrix terms, with a consequent halving of the orders of the determinants involved.