Tridiagonal normal forms for orthogonal similarity classes of symmetric matrices Original Research Article

Let F be an algebraically closed field of characteristic different from 2. Define the orthogonal group, On(F), as the group of n by n matrices X over F such that XX′=In, where X′ is the transpose of X and In the identity matrix. We show that every n by n symmetric matrix over F is orthogonally similar to a tridiagonal symmetric matrix.If further the characteristic is 0, we construct the tridiagonal normal form for the On(F)-similarity classes of symmetric matrices. We point out that, in this case, the known normal forms (as presented in the well known book by Gantmacher) are not tridiagonal.