Tridiagonal matrices
Let \(A\) be an \(n \times n\) symmetric matrix with elements in a principal ideal domain \(R\). The author proves that if \(\text{det} A = d \neq 0\) then \(A\) is congruent to a tridiagonal matrix \[ \left[ \begin{matrix} c_ 1 & d_ 1 & & & \;\\ d_ 1 & c_ 2 & d_ 2 & & \;\\ & \ddots & \ddots & \ddots & \;\\ & & d_{n-2} & c_{n-1} & d_{n-1} \\ & & & d_{n-1} & c_ n \end{matrix} \right] \] where \(d_ i | d\), \(1 \leq i \leq n-2\). He also proves that if \(A\) is unimodular, then the off diagonal elements may be chosen so that \(d_ i=1\), \(1 \leq i \leq n - 2\).
scientific article; zbMATH DE number 614169