Root distribution for the ellipse
Eugene Walach, E. Zeheb · IEEE Transactions on Automatic Control · 1982
It is shown that the number of characteristic roots of a given matrix (or polynomial) inside, outside, and on the boundary, of a given ellipse, can be determined by a common half-plane test (e.g., Routh test). The test should be performed on a polynomial which is received from the original characteristic polynomial by a simple transformation. Root exclusion and root clustering criteria are included as special cases.