Nearest southeast submatrix that makes two prescribed eigenvalues
Abstract
Given four complex matrices $A$, $B$, $C$ and $D$ where $A \in
\mathbb{C}^{n\times n}$ and $D \in \mathbb{C}^{m\times m} $ and
given two distinct arbitrary complex numbers $ \lambda_{1} $ and $
\lambda_{2} $, so that they are not eigenvalues
of the matrix $A$,
we find a nearest matrix from the set of
matrices $ X \in \mathbb{C}^{m \times m } $ to matrix $D$ (with
respect to spectral norm) such that the matrix
$\begin{pmatrix}
A & B \\
C & X
\end{pmatrix} $
has two prescribed eigenvalues $ \lambda_{1} $ and $ \lambda_{2}$.
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