Poncelet–Darboux, Kippenhahn, and Szegő: Interactions between projective geometry, matrices and orthogonal polynomials

On foci of ellipses inscribed in cyclic polygons

“…The set of all n × n matrices satisfying properties (i-iii) is precisely the set S n that we described in the Introduction. As we stated there, it is known that the numerical range of a matrix in S n is the convex hull of an algebraic curve of class n and has the (n + 1)-Poncelet property, meaning that every point on ∂W (A) is a point of tangency for a convex (n + 1)-gon that is circumscribed about W (A) and inscribed in T (see [13] for details).…”

“…There are several canonical forms of matrices from the class S n (see [13,Section 2.3]) and the one that we will use is that of a cutoff CMV matrix. To define a CMV matrix, first define a sequence…”

“…This phenomenon can be reformulated as saying that ∂W (G (n) ) is the envelope of the family of circumscribing (n + 1)-gons. The precise definition of an envelope and its relationship to numerical ranges is complicated, so we will restrict our attention only to the most relevant facts and refer the reader to [13] for details. The envelope is most easily understood by means of a dual curve.…”

“…One simplification of our problem comes from the fact that instead of finding a matrix in S n−1 with the desired properties, it suffices to find just the eigenvalues of that matrix. This is because numerical ranges are preserved by unitary conjugation and every matrix in S n−1 is unitarily equivalent to a canonical form called a cutoff CMV matrix (see [13,14]). Such matrices are an important part of the theory of orthogonal polynomials on the unit circle and will play an essential role in our analysis.…”

“…The next section is a brief review of the background, notation, and terminology that will be relevant to the remainder of the paper. Many of these topics are discussed in much greater detail in [3,13], and a thorough introduction to orthogonal polynomials on the unit circle can be found in [19,20].…”