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In my statistics textbook, it says the special correlation structure $\text{Cov}(X_i, X_k)=\sqrt{\sigma_{ii}\sigma_{kk}} \rho$, or $\text{Corr}(X_i, X_k)=\rho$, all $i \neq k$, is one important structure in which the eigenvalues of $\boldsymbol { \Sigma}$ are not distinct.

My question: I don't know why the correlation matrix $$\boldsymbol{\rho}=\begin{bmatrix}1&\rho & \cdots &\rho\\\rho&1&\cdots &\rho \\ \vdots & \vdots & \ddots & \vdots \\ \rho &\rho &\cdots &1 \\\end{bmatrix}$$

can imply the eigenvalues are not distinct.

Mariana
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  • https://math.stackexchange.com/questions/2306356/are-eigenvalues-of-a-and-b-the-same-if-b-is-a-permutation-of-a – Eric Towers Nov 13 '21 at 01:10
  • @EricTowers I don't think these are the same questions. – Mariana Nov 13 '21 at 01:16
  • Your matrix is similar to is eigendecomposition. Permutation of the eigenvalue diagonal matrix in that decomposition corresponds to symmetric (rows and columns) permutation of the matrix you give. Since symmetric permutation leaves your matrix unchanged, it must leave the eigenvalue diagonal matrix unchanged. Consequently, all the eigenvalues of your matrix are the same. – Eric Towers Nov 13 '21 at 01:18
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    There are two distinct eigenvalues of this correlation matrix: https://math.stackexchange.com/q/689111/321264, https://math.stackexchange.com/q/55165/321264 – StubbornAtom Nov 13 '21 at 06:26
  • The question is NOT asking the eigenvalues of the correlation matrix. The question is about the eigenvalues of the $\Sigma$ – Mariana Nov 20 '21 at 00:33

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