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Show that if $T$ is a symmetric tridiagonal matrix and an eigenvalue has multiplicity $k$, then at least $k−1$ subdiagonal elements of $T$ are zero.

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Hint. If $T$ has $r$ zeros on its sub/super-diagonal, we can split $T$ into a direct sum of $r+1$ diagonal blocks, each with entrywise nonzero sub/super-diagonal. So, it suffices to show that every such diagonal sub-block has a spectrum of distinct eigenvalues.

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