I'm looking at $N\times N$ matrices $M_N$ of the form $$M_4=\begin{pmatrix}1 & a & a^2 & a^3 \\ a & 1 & a & a^2 \\ a^2 & a & 1 & a \\ a^3 & a^2 & a & 1\end{pmatrix},$$ where $a$ is a complex number of unit modulus. I'm particularly interested in large matrices with the property $a^N=1$. I've come up empty in several attempts trying to find eigenvalues or eigenvectors of this matrix analytically for sizes larger than 4, but the fact that this matrix only has one parameter and its close relation to Toeplitz matrices makes me hope someone has studied these things before.
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- 1$\begingroup$ See the links in math.stackexchange.com/q/3109822/321264. $\endgroup$StubbornAtom– StubbornAtom2019-03-03 12:16:26 +00:00Commented Mar 3, 2019 at 12:16
- $\begingroup$ Thanks a lot @StubbornAtom! Unfortunately all links discuss real $a$ with $|a|<1$. But I'll have a look, perhaps I can use some of the conclusions. $\endgroup$Daniel– Daniel2019-03-03 18:04:05 +00:00Commented Mar 3, 2019 at 18:04
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