Combining triple diagonal forms
Let \(R\) be a principal ideal domain of characteristic not equal to 2. Two matrices \(A,C \in M_n(R)\) are said to be congruent if \(C=P^TAP\) for some \(P \in GL_n(R)\). It is known (Jacobi) that a symmetric matrix \(A \in M_n(R)\) is congruent to a tridiagonal matrix. This form was strengthened by some authors in different ways. In this paper the author works towards combining these refinements. Specifically, he proves the following main result: Let \(R\) be a Euclidean domain of characteristic not equal to 2, and let \(A\) be a symmetric matrix in \(M_n(R)\) such that \(d= \det{A} \neq 0\). Then, \(A\) is congruent to either a tridiagonal matrix \[ C=\left( \begin{matrix} c_1 & d_1 & 0 & \ldots & 0 & 0 \\ d_1 & c_2 & d_2 & \ldots & 0 & 0 \\ 0 & d_2 & c_3 & \dots & 0 & 0 \\ \vdots & \vdots &\vdots & & \vdots & \vdots \\ 0 & 0 & 0 & \dots & c_{n-1}& d_{n-1} \\ 0 & 0 & 0 & \dots & d_{n-1}& c_n \end{matrix} \right), \] where both the following hold: {\parindent=6.5mm \begin{itemize}\item[(i)] \(|d_1| \leq |d_2| \leq \cdots \leq |d_{n-1}|\), \item[(ii)] \(d_i | d\) for \(i \leq n-4\), \end{itemize}} or a block diagonal matrix with each block satisfying this condition.
scientific article; zbMATH DE number 5833983