Hi everyone,
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 real with exactly n eigenvalues positive and the other n eigenvalues 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.
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 real with exactly n eigenvalues positive and the other n eigenvalues 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.
0 commentaires:
Enregistrer un commentaire