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I have a square matrix [tex]J \in \mathbb{C}^{2n \times 2n}[/tex] where,[tex]J=\left(\begin{array}{cc}A&B\\\bar{B}&\bar{A}\end{array}\right)[/tex][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 any J with the above structure. But suppose it holds, 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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# Eigenvalue separation of a Block Matrix with a special structure

Can you offer guidance or do you also need help?

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