Eigenvalues of matrix $(A)_{nm} = e^{i\phi |n-m|}$

eigenvalues-eigenvectorsmatricessymmetric matricestoeplitz-matrices

I'm considering $(N+1\times N+1)$-matrices of the form

\begin{equation}
\newcommand{\iu}{\mathrm{i}}
\newcommand{\euler}{\mathrm{e}}
A = \begin{pmatrix}
1 & \euler^{\iu\phi} & \euler^{\iu2\phi} &\dots& \euler^{\iu N\phi} \\
\euler^{\iu\phi} & 1 & \euler^{\iu\phi} & \dots & \euler^{\iu (N-1)\phi} \\
\euler^{\iu2\phi} & \euler^{\iu\phi} & 1 & & \vdots \\
\vdots & \vdots& &\ddots& \euler^{\iu\phi}\\
\euler^{\iu N\phi} & \euler^{\iu(N-1)\phi} & \dots &\euler^{\iu\phi}& 1
\end{pmatrix},
\end{equation}
i.e., matrices $(A)_{nm} = \euler^{\iu\phi|n-m|}$.

I want to have a (if possible closed form) expression for the eigenvalues. Mathematica readily finds them, even for $N = 30$. Also the structure of these matrices is relatively simple. Thus I assume there are probably known results for something like the eigenvalues and eigenvectors of $A$. However, I wasn't able to find my matrix in the list of matrices.

My questions are:

  • For $\phi \in \mathbb{R}$ are these matrices called and what are, for a general $N$, the eigenvalues of $A$? This question is likely related to this question (thanks to omnomnomnom)
  • If there is no closed form, is there are an approximate expression for the eigenvalues and eigenvectors for $N\to\infty$?

Best Answer

Following answer is probably as close to a solution as one can get within reasonable effort. Much of what I'm about to write is based on following two questions and answers therein:

Diagonalizing a [...] Symmetric Toeplitz matrix

[...] determinant of a tridiagonal matrix with constant diagonals

Part 1 - Inverse of $A$

$A$ possesses an inverse, given by

\begin{equation} (A^{-1})_{nm} = \frac{1}{1-e^{2i \phi}} \begin{cases} 1 & n = m = 1, N \\ 1+e^{i\phi} & n = m \\ - e^{i\phi} & |n-m| = 1 \\ 0 & \text{otherwise} \end{cases}, \end{equation} i.e., the inverse of $A$ is a tridigonal matrix with near constant diagonals, except for $(A^{-1})_{1,1}$ and $(A^{-1})_{N,N}$.

Part 2 - Characteristic Polynomial of $A^{-1}$ For the calculation of the characteristic Polynomial

$$\chi(\lambda) = \det\big(A^{-1} - \lambda\big)$$

it's useful to consider the substitution

$$\lambda = \frac{1}{1-e^{2 i \phi}} \mu + \frac{1+e^{2i\phi}}{1-e^{2i\phi}}.$$

With that, we have $\det(A^{-1} - \lambda) = 0 \Leftrightarrow \det(\tilde{A}^{-1} - \mu) = 0$ with

\begin{equation} \tilde{A}^{-1} = \begin{cases} -e^{2i\phi} & n = m = 1,N \\ -e^{ i\phi} & |n-m| = 1 \\ 0 & \text{otherwise} \end{cases} \end{equation}

Now, by defining the auxilliary $N\times N$ matrix $B_N$ which contains $-e^{i\phi}$ on the first of diagonals and $0$ on all other elements (i.e. $(B_N)_{n,m} = (\tilde{A}^{-1})_{n,m}$ expect for $(B_N)_{1,1} = (B_N)_{N,N} = 0$), one derives

\begin{equation} \det(\tilde{A}^{-1} - \mu) = (e^{2i\phi} - \mu)^2 \det(B_{N-2}) + 2 e^{2i\phi} (e^{2i\phi} - \mu) \det(B_{N-3}) + e^{4i\phi} \det(B_{N-4}). \end{equation}

For $\det(B_N)$ one can derive a recursion relation, like in the second linked question, which gives $\det(B_N)$ in terms of Chebyshev polynomials. For $\mu^2 \ne 4 e^{2i\phi}$ and after one further substitution $\tilde{\mu} = \mu e^{-i\phi}$ one finds the characteristic Polynomial

\begin{align} &\det(\tilde{A}^{-1} - \mu) = 0 \Leftrightarrow \\ &\color{blue}{(e^{i\phi} - \tilde{\mu})^2 U_{N-2}(\tilde{\mu}/2) - 2(e^{i\phi} - \tilde{\mu}) U_{N-3}(\tilde{\mu}/2) + U_{N-4}(\tilde{\mu}/2) = 0}, \end{align} where $U_N$ is the Chebyshev Polynomial of the second kind.

Part 3 - short discussion

With the knowledge of the solutions of the characteristic equation for $\tilde{\mu}$ on directly finds the the eigenvalues $\lambda$ of $A^{-1}$ and, with that, for $A$. $U_N(x)$ is a polynomial of degree $N$ with $N$ distinct roots $x_k = \cos(\pi k/(N+1))$ with $k=1,\dots,N$. One may hope, that these are sufficiently close to the actual roots and do pertubation theory on those roots (e.g. by using Newton's method to calculate corrections to $x_k$). However, we are calculating the eigenvalues of $A^{-1}$ and some of the (approximative) eigenvalues $x_k$ lie close to $0$ (namely for $k$ close to $N/2$). Thus, small corrections to those, mean enormous corrections to the resulting approximative eigenvalue of $A$. Therefore finding approximative eigenvalues with the above stated "algorithm" is not well suited.

Finally, there is one more option, to get to a "simpler" eigenvalue equation for $A^{-1}$. By substituting once more $\tilde{\mu} = \nu + \nu^{-1}$ the Chebyshev Polynomials reduce to $U_N(\tilde{\mu}/2) = \nu^{N-1}-\nu^{-N+1}$. After rearranging some powers one ends up with

\begin{align} &\det(\tilde{A}^{-1} - \mu) = 0 \Leftrightarrow \\ &\nu^{2N}(e^{i\phi} + \nu)^2 - (1 + e^{i\phi}\nu)^2 = 0. \end{align}

Notice, this is now a polynomial of degree $2N+2$, however $\nu = \pm 1$ is a solution, which we already excluded, since we assumed $\mu^2 \ne 4e^{2i\phi}$ and all other roots come in pairs: if $\nu_k$ is a solution, so is $\nu_k^{-1}$. However, they provide the same $\tilde{\mu}$ after resubstitution. All in all, while this polynomial equation is much more compact, it didn't helped me find a closed form solution or an good approximation of the original problem (this is, the eigenvalues of $A$).

Related Question