Computing symmetric rank-revealing decompositions via triangular factorization (Q2784356)

The authors present a family of algorithms for computing symmetric rank-revealing \(VSV\) decompositions based on a triangular factorisation. Such decompositions include a symmetric matrix \(S\) revealing the numerical rank via three blocks of small norm and an orthogonal matrix \(V\) whose columns span approximations to the numerical range and the null space. It is shown that for semidefinite matrices the \(VSV\) decomposition should be determined through \(ULV\) decomposition, while through \(URV\) involving hypernormal rotations for indefinite ones. Numerical examples illustrate the feasibility of the approach.