UB-matrices and conditions for Poncelet polygon to be closed
An \(n\)-by-\(n\) matrix \(A\) is said to be UB-matrix if \(A= V^*UV\), where \(U\) is an \((n+1)\)-by-\((n+1)\) unitary matrix with distinct eigenvalues, and \(V\) is an \((n+1)\)-by-\(n\) matrix with \(V^* V= I_n\) such that the inner product of any nonzero vector in the orthogonal complement of the range space of \(V\) with any eigenvector of \(U\) is nonzero. Such matrices can also be characterized as the ones \(A\) with \(\|A\|\leq 1\), eigenvalues in the open unit disc \(\mathbb{D}\) and rank \((I_n- A^*A)= 1\). In recent years, it was proved by the present author [ibid. 281, No. 1-3, 59-85 (1998; Zbl 0936.15024)] and, independently, by \textit{H.-L. Gau} and the reviewer [Linear Multilinear Algebra 45, No. 1, 49-73 (1998; Zbl 0918.15008)] that the numerical range \(W(A)\) of such a matrix \(A\) has the \((n+1)\)-Poncelet property, namely, for every point \(\lambda\) on the unit circle \(\partial\mathbb{D}\) there exists an \((n+1)\)-gon which is inscribed in \(\partial\mathbb{D}\), circumscribed about the boundary \(\partial W(A)\) and has \(\lambda\) as a vertex. In this case, the \((n+1)\)-gon has the eigenvalues of \(U\) as its vertices. For a UB-matrix \(A\) and any real \(\theta\), let the eigenvalues of the Hermitian matrix \(\cos\theta\text{ Re }A+ \sin\theta\text{ Im }A\) be arranged in descending order: \(\lambda_1(\theta)>\cdots> \lambda_n(\theta)\) (they are known to be distinct). Each of these defines a closed curve \(K_j\), \(1\leq j\leq n\), via \[ \begin{aligned} x(\theta) &= \lambda_j(\theta)\cos\theta- \lambda_j'(\theta)\sin\theta,\\ y(\theta) &= \lambda_j(\theta)\sin\theta+ \lambda_j'(\theta)\cos\theta.\end{aligned} \] Since \(K_j\) and \(K_{n-j+1}\) coincide, these yield \([(n+1)/2]\) nested curves \(K_1,\dots, K_{[(n+1)/2]}\). They form a package of Poncelet curves because \(K_1\) is exactly the boundary of \(W(A)\) while \(K_j\), \(2\leq j\leq [(n+ 1)/2]\), is the envelope for the diagonals of the \((n+1)\)-gon which connect its every \(j\)th vertices. A natural question arises whether this is the only way to produce such packages. In this paper, the author considers the resultant of two polynomials associated with the eigenvalues of a UB-matrix and analyzes the density function of a measure connected with the enveloping property to obtain a sufficiency condition for the generation of a package of Poncelet curves (Theorem 1). This is used to construct in Example 1 a package which does not arise from any UB-matrix, thus refuting a previous conjecture. Another application of this condition is to give a characterization of a package of Poncelet curves among a given package of nested ellipses. Using this, the author gives all the possible locations for the foci of ellipses in such a package. Such results are alternatives to the classical Cayley criterion for an \((n+1)\)-gon to interscribe between two given ellipses. Although slightly difficult to follow, this paper contains many interesting results which are worthy of detailed explorations.
scientific article; zbMATH DE number 1880031