# Diagonalization of the following Toeplitz complex symmetric matrix

There is a Toeplitz matrix of the following form:

$$$$M = \begin{pmatrix} 1 & e^{i\phi} & e^{2i\phi} & \ldots & e^{(N-1)i\phi} \\ e^{i\phi} & 1 & e^{i\phi} & \ldots & e^{(N-2)i\phi} \\ e^{2i\phi} & e^{i\phi} & 1 & \ldots & e^{(N-3)i\phi} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ e^{(N-1)i\phi} & e^{(N-2)i\phi} & e^{(N-3)i\phi} & \ldots & 1 \end{pmatrix}.$$$$

where $$\phi$$ is just some real number. This is a complex symmetric Toeplitz matrix. There is no general algorithm to diagonalize it analytically, and I am not absolutely sure that this is possible. Right now I am in the process of obtaining the eigenvalues (or at least a simple form of the eigenvalue equation).

However, what I do know is the answer in the case of $$\phi=0$$, then it is just a matrix of all $$1$$'s, which is a circulant type, and can be diagonalized by the DFT matrix. So, I wonder, may be there is some sort of simple transformation between the $$\phi=0$$, and $$\phi \ne 0$$ cases so that I can find the eigenvectors without explicitly finding the Kernel for each $$M-\lambda_jI$$?

### Background

Matrix $$M$$ describes a certain periodic quantum-mechanical system. Physically, it is somewhat similar to what is called a photonic crystal, where each subsystem interacts with others by means of plane waves (hence the constant amplitude of all entries, and an acquired multiple of $$\phi$$ in phase).

Update 23.03.2021 So, basically, I was able to obtain an explicit form of the eigenvalue equation: $$$$\det \left(M - \lambda I \right) = \dfrac{\left( \beta_+^N - \beta_-^N \right) (1-\lambda) + (\beta_-^N \beta_+ - \beta_+^N \beta_-)}{\beta_+-\beta_-} = 0, \\ \beta_{\pm} = \frac{1}{2} \left( 1-\lambda - e^{i2\phi}(1+\lambda) \pm \sqrt{ (1-\lambda - e^{i2\phi} (1+\lambda))^2 - 4 e^{i2\phi} \lambda^2 } \right),$$$$ which is... fairly complicated. I believe this can not be solved explicitly, and, probably, the eigenvectors can not be expressed exactly. However, there is also a way to solve it by using a physical rationale, mb I will share some materials on that later.

• The matrix of all ones can be diagonalized by plenty of matrices, since it is highly degenerate - I am not sure that is useful insight. Commented Mar 22, 2021 at 15:36
• Hi again, Redrigo de Azevedo! This matrix describes a certain periodic quantum-mechanical system, physically it is somewhat similar to what is called a photonic crystal, where each subsystem interacts with others by means of plane waves (hence the constant amplitude of all entries, and an acquired multiple of $\phi$ in phase). Commented Mar 22, 2021 at 15:45
• @Igor, indeed, for $\phi=0$ there is 1 mode with an eigenvalue of $\lambda_1=N$ (sometimes called in physics a superradiant mode), while $N-1$ of others are degenerate and have $\lambda_j=0$ (called subradiant, as they have $0$ emission rate). Commented Mar 22, 2021 at 15:50
• I would write $e^{ik\phi}$ instead. Imaginary unit comes first. Commented Mar 22, 2021 at 16:24

Computing the eigensystem with Mathematica indicates that there is nothing particularly enlightening going on. Perhaps in the case where $$\exp(i \theta)$$ is a root of unity, something may be doable, but in general...