A Solution of Inverse Eigenvalue Problems for Unitary Hessenberg Matrices

F. Li and
L. Lin

Numer. Math. J. Chinese Univ. (English Ser.)(English Ser.) 16 (2007), pp. 131-139

Let $H \in {\mathbb C}^{n\times n}$ be an $n
\times n $ unitary upper Hessenberg matrix whose subdiagonal
elements are all positive. Partition $H$ as
\begin{equation}\label{eq1}
H= \left [ \begin{array}{cc}
H_{11}&H_{12}\\
H_{21}&H_{22}
\end{array} \right ],
\end{equation}
where $H_{11}$ is its $k\times k$ leading principal submatrix;
$H_{22}$ is the complementary matrix of $H_{11}$. In this paper, $H
$ is constructed uniquely when its eigenvalues and the eigenvalues
of $\widehat{H}_{11}$ and $\widehat{H}_{22}$ are known. Here
$\widehat{H}_{11}$ and $\widehat{H}_{22}$ are rank-one modifications
of $ H_{11} $ and $H_{22}$ respectively.