UB-matrices and conditions for Poncelet polygon to be closed
type: Item
instance of: scholarly article
title: UB-matrices and conditions for Poncelet polygon to be closed
zbMATH Open document ID: 1028.15025
DOI: 10.1016/S0024-3795(02)00447-0
author: Boris Mirman
published in: Linear Algebra and its Applications
publication date: 2003-03-11T00:00:00Z
review text: 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…
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 1880031
zbMATH Keywords: unitary dilation
zbMATH Keywords: Poncelet's porism
zbMATH Keywords: UB-matrix
zbMATH Keywords: eigenvector
zbMATH Keywords: eigenvalues
zbMATH Keywords: numerical range
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1880031