= Positive diagonal symmetrization of a tridiagonal matrix
{title2=$d_{i+1}/d_i=\sqrt{c_i/b_i}$}
Let a real <tridiagonal matrix> have diagonal entries $a_i$, positive upper entries $b_i$, and positive lower entries $c_i$. Set $d_1=1$, $d_{i+1}=d_i\sqrt{c_i/b_i}$ and $D=\operatorname{diag}(d_i)$. Then $D^{-1}AD$ is a <symmetric matrix> with diagonal $a_i$ and adjacent entries $\sqrt{b_ic_i}$. Indeed its two adjacent entries are $b_id_{i+1}/d_i$ and $c_id_i/d_{i+1}$, which are equal by construction. All other zeros are preserved. This <matrix similarity> implies real <eigenvalues> and <diagonalizability>, by the <spectral theorem for real symmetric matrices>. No sign assumption on $a_i$ is needed. In the weighted formulation of <positive diagonal symmetrization of a matrix>, the positive <diagonal matrix> is $W=D^{-2}$.
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