Absolutely indecomposable symmetric matrices
It is well known that every real symmetric matrix can be diagonalized by means of conjugation with an orthogonal matrix. But this is not the case for matrices over arbitrary fields. The authors focus on the more general problem, i.e., the decomposability of a given symmetric matrix into smaller blocks orthogonally. A symmetric matrix \(A\) is called absolutely indecomposable if it is indecomposable over every extension of the base field. This can only happen if the base field \(K\) is a non-formally real field. Assume that \(K\) is such a field of level \(s\) (i.e. \(s\) is the smallest integer such that \(-1\) is a sum of \(s\) squares in \(K\)), then it is proved that in Mat\(_n(K)\) there exist symmetric absolutely indecomposable matrices iff \(n>s\) and \(n\equiv 0\pmod {2s}\) or \(n\equiv \pm 1\pmod {2s}\). The proofs yield an explicit description of such matrices. On the other hand, if \(n\leq s\), it is shown that every symmetric matrix can be diagonalized orthogonally over some extension field of \(K\).
scientific article; zbMATH DE number 1839049