# Property of symmetric tridiagonal matrices with an eigenvalue of multiplicity $m$

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.

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.