Continuous bordering of matrices and continuous matrix decompositions (Q1893492)

This paper studies the problems in continuous bordering of matrices, which are often used to compute manifolds of fold points or symmetry breaking bifurcation points. It has been concluded that for \(n \times n\) matrices, with \(n \geq 2\) and rank \(\geq n - 1\), there are no continuous vector fields that will preserve the regularity of their bordered matrices. It is also shown that there is no singular value decomposition (SVD) of a matrix depending continuously on itself, and that such branching steps in algorithms for SVD and similarly for eigenvalue decompositions are mathematically unavoidable.