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When do symmetric matrices with rotated rows have repeated eigenvalues? By rotated rows I mean each row is a rotation of a previous row.

In $\mathbb R^n$, the matrix $S$ where $$S_{ij} = \begin{cases}1 &\text{ if } i = j\\ \frac 1 2 &\text{ otherwise}.\end{cases}$$ has $n - 1$ repeated eigenvalues, as shown by computation here.

Does this generalize? Can we state conditions where matrices that resemble $S$ have $k$ repeated eigenvalues?

See also When does a symmetric matrix have repeated eigenvalues? , which is related, but does not address matrices of this nature.

SRobertJames
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  • What are "rotated rows"? – Ted Shifrin Jan 21 '24 at 19:59
  • all matrices of that kind have $n-1$ repeated eigenvalues since they can be written as $cI + E$ where $I$ is the identity, $c$ is a scalar and $E$ is a rank one matrix – Exodd Jan 21 '24 at 19:59
  • @TedShifrin Updated – SRobertJames Jan 21 '24 at 20:02
  • see for example https://math.stackexchange.com/questions/689111/find-the-eigenvalues-of-a-matrix-with-ones-in-the-diagonal-and-all-the-other-el – Exodd Jan 21 '24 at 20:03
  • If you mean that the first row may be any vector, then "matrices with rotated rows" are well-known and they are called Circulant Matrices, whose eigenvalues are easy to compute, but they are almost always distinct https://en.wikipedia.org/wiki/Circulant_matrix – Exodd Jan 21 '24 at 20:09
  • @Exodd You have some valuable points. Would you consider assembling and expanding them into an answer? – SRobertJames Jan 21 '24 at 20:12

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