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Eigenvalue separation of a Block Matrix with a special structure

Sam1984

New member
Feb 27, 2014
1
Hi everyone,

I have a square matrix [tex]J \in \mathbb{C}^{2n \times 2n}[/tex] where,

$J=\left(\begin{array}{cc}A&B\\\bar{B}&\bar{A}\end{array}\right)$

[tex]A \in \mathbb{C}^{n \times n}[/tex] and its conjugate [tex]\bar{A}[/tex] are diagonal.Assume the submatrices [tex]A,B \in \mathbb{C}^{n \times n}[/tex] are constructed in a way that all $2n$ eigenvalues are either real with exactly n eigenvalues positive and the other n eigenvalues negative or if some eigenvalues are complex the real part of these $2n$ eigenvalues are half positive half negative. Notice that this property does not hold in general for every $J$ with the above structure. But suppose it holds for a set of [tex]A,B \in \mathbb{C}^{n \times n}[/tex], then how can we form a new $n$ by $n$ matrix based on the submatrices [tex]A,B \in \mathbb{C}^{n \times n}[/tex] which has only the positive eigenvalues of $J$ as its set of eigenvalues.

Any help would be greatly appreciated.
 
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