Eigenvalue separation of a Block Matrix with a special structure

[Sam1984]

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.

 

[jvicens]

Hi there,

Thank you for sharing your interesting question. It seems like you are looking for a way to construct a new $n$ by $n$ matrix with only the positive eigenvalues of $J$ as its set of eigenvalues. This can be achieved by using the Schur decomposition of the matrix $J$, which allows us to transform it into an upper triangular matrix with its eigenvalues on the diagonal.

In this case, we can use the Schur decomposition to obtain a new matrix $T$ such that $J=UTU^*$, where $U$ is a unitary matrix and $T$ is an upper triangular matrix. The eigenvalues of $J$ will be the same as the diagonal elements of $T$. Since $T$ is upper triangular, the positive eigenvalues of $J$ will be located on the diagonal of $T$.

To construct a new $n$ by $n$ matrix with only the positive eigenvalues of $J$, we can simply take the top left $n$ by $n$ submatrix of $T$, which will contain only the positive eigenvalues. This new matrix will have the desired property that all its eigenvalues are positive.

I hope this helps. Let me know if you have any further questions. Good luck with your research!