Tridiagonal normal forms for orthogonal similarity classes of symmetric matrices
Considering the orthogonal group of matrices of order \(n\) over an algebraically closed field of characteristic different from 2, this paper presents a study of similarity action of this group on the space of all symmetric matrices of order \(n\) over the mentioned field. It is shown that each group orbit in the space of all symmetric matrices has a representative that is the direct sum of tridiagonal blocks, whose super- and subdiagonal entries are 1, each block being similar to a single Jordan block. If the field has characteristic 0, then the mentioned tridiagonal blocks can be replaced with new tridiagonal blocks which can be (unique) symmetrized Jordan blocks, this allowing to obtain a new type of normal form for symmetric matrices. In this case, the known normal forms (according to the classical book of \textit{F. R. Gantmacher}, The theory of matrices (1953; Zbl 0050.24804)) are not tridiagonal. Finally, four open problems are formulated.
scientific article; zbMATH DE number 2083493