Let $T$ be the $2 N \times 2 N$ matrix defined by $$ T = \begin{pmatrix} A && B\\ -B && -A^* \end{pmatrix} $$ where $*$ is entry wise complex conjugation, $A$ is a Hermitian $N \times N$ tridiagonal Toeplitz matrix $$ A = \begin{pmatrix} a & \alpha& 0 & \dots & 0\\ \alpha^* & a & \alpha & \vdots & \vdots\\ 0 & \alpha^* & \ddots & \ddots & \vdots\\ \vdots & \ddots & \ddots & a & \alpha\\ 0 & \dots & \dots & \alpha^* & a \end{pmatrix} $$ with $a$ real and $\alpha$ in general complex and $B$ is a matrix proportional to the identity $$ B = b Id_{N \times N} $$ where $b$ is just a real number. I want to find the eigenvalues of eigenvectors of $T$. Here's what I have so far.

Finding an eigenvalue $\lambda$ is equivalent to solving the difference equation $$ D \begin{pmatrix} x_{j-1} \\ y_{j-1} \end{pmatrix} + (E-\lambda) \begin{pmatrix} x_{j} \\ y_{j} \end{pmatrix} + D \begin{pmatrix} x_{j+1} \\ y_{j+1} \end{pmatrix} = \begin{pmatrix} 0\\ 0 \end{pmatrix} $$ with boundary conditions $x_0 = y_0 = x_{N+1} = y_{N+1}$where $$ D = \begin{pmatrix} \alpha & 0 \\ 0 & \alpha \end{pmatrix} \:\:\: E = \begin{pmatrix} a & b \\ -b & -a \end{pmatrix} $$ This can be simply solve by setting $x_j = sin(k j) x, y_j = \sin(k j) y$ with $k = \frac{\pi n}{N+1}$ , $n = {1, \dots, N}$, and (x,y) satisfy $$ (2 \cos(k)D+(E-\lambda)\big) \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 0 \\ 0 \end{pmatrix} $$ So we have all our eigenvalues and eigenvectors. This really just comes down to the fact that for $\alpha$ real, our block matrices $A$ and $B$ (which is of course just proportional to the identity) commute with each other.\

Now when $\alpha$ is complex, the same procedure as above fails. So what I did next was at least try and find the eigenvalues $\lambda$ first to see if that would help. Note that our the determinant of our matrix $T$ has the nice property that $$ \det(T-\lambda) = \det(-(A-\lambda)(A^*+\lambda)+b Id_{N \times N} = 0 ) $$ So now we've reduced the problem of finding the eigenvalues to a problem of a finding the zero of the determinant of a $2N \times 2 N$ matrix. Now this matrix is a almost a Toeplitz matrix, so I was wondering if anyone knew if it is possible to find the zeroes of this matrix. Or even better, the eigenvectors of the matrix $T$. Thanks!

  • $\begingroup$ Try finding rank of the matrix. $\endgroup$ – G_0_pi_i_e Sep 12 '17 at 2:36

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