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[Community Question] Linear-algebra: A computation of a zero eigenvalue

One of our user asked:

Let $d\geq 3$. Consider the $(2d-2) \times (2d-2)$ matrix with $a_{ii} = 1$, $a_{ij} = -\frac{1}{3}$ if $|i-j|=1$ and $a_{ij} = \frac{1}{3}$ otherwise.

Prove that its smallest eigenvalue is $0$ and its multiplicity is $d-2$.

It might be helpful to decompose the matrix as $AA^T$ (as far as I know, this has to be true) but I don't know how to do this effectively either - not much experience in Cholesky factorization.

Any help appreciated!


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