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According to Wikipedia, a tridiagonal matrix is a band matrix that has nonzero elements only on the main diagonal, the subdiagonal and the supradiagonal. To my understanding, in a tridiagonal stochastic matrix, from state $i$, we could always reach another state $j$ (imagining transitioning to neighboring states in each step). Therefore, I thought that a tridiagonal stochastic matrix is always irreducible. However, in this paper, it is stated that

A tridiagonal matrix ($a_{ij}$) is irreducible if and only if $a_{ij}a_{ji} \neq 0$ for all $j = i + 1$.

This statement confused me because I thought that $a_{ij} a_{ji} \neq 0$ is always true for a tridiagonal matrix. I'm wondering if I'm missing anything here.

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    $\begingroup$ Most of the time, tridiagonal means that the only non-zero entries are in those three diagonals but not that the entries of those diagonals are all non-zero. $\endgroup$ Oct 30, 2022 at 7:29
  • $\begingroup$ Ah I see. I misread the definition in Wikipedia. Thanks for pointing that out! $\endgroup$
    – Jack
    Oct 30, 2022 at 7:30

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As @Mariano Suárez-Álvarez pointed out in the comment section, it is not required in a tridiagonal matrix that all entries of the main diagonal, subdiagonal and the supradiagonal be non-zero, which necessitates the condition that $a_{ij}a_{ji}\neq 0$ for a tridiagonal stochastic matrix to be irreducible.

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