# Eigenvalue separation of a Block Matrix with a special structure

#### Sam1984

##### New member
Hi everyone,

I have a square matrix $$J \in \mathbb{C}^{2n \times 2n}$$ where,

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

$$A \in \mathbb{C}^{n \times n}$$ and its conjugate $$\bar{A}$$ are diagonal.Assume the submatrices $$A,B \in \mathbb{C}^{n \times n}$$ 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 $$A,B \in \mathbb{C}^{n \times n}$$, then how can we form a new $n$ by $n$ matrix based on the submatrices $$A,B \in \mathbb{C}^{n \times n}$$ which has only the positive eigenvalues of $J$ as its set of eigenvalues.

Any help would be greatly appreciated.

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