The article is devoted to different aspects of the question: ”What can be done with a complex-valued matrix by a low rank perturbation?” From the works of Thompson [15] we know how the Jordan normal form can be changed by a rank k perturbation, see Theorem 2. Particulary, it follows that one can do everything with a geometrically simple spectrum by a rank 1 perturbation, see Corollary 1. But th...

Given a square matrix A, a Brauer’s theorem [Limits for the characteristic roots of matrices IV: Applications to stochastic matrices, Duke Math. J. 19 (1952), 75-91] shows how to modify one single eigenvalue of A via a rank-one perturbation without changing any of the remaining eigenvalues. Older and newer results can be considered in the framework of the above theorem. In this paper, we presen...

Many idealized problems in signal processing, machine learning and statistics can be reduced to the problem of finding the symmetric canonical decomposition of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing inspiration from the matrix case, the successive rank-one approximations (SROA) scheme has been proposed and shown to yield this tensor decomposition exactly, an...

Many idealized problems in signal processing, machine learning and statistics can be reduced to the problem of finding the symmetric canonical decomposition of an underlying symmetric and orthogonally decomposable (SOD) tensor. Drawing inspiration from the matrix case, the successive rank-one approximations (SROA) scheme has been proposed and shown to yield this tensor decomposition exactly, an...

A conjecture from [3] concerning the location of eigenvalues rank one perturbations singular M-matrices is shown to be false in dimension four and higher, but true for two, as well three with an additional condition on perturbation.

We say that a square complex matrix is condensable if it can be reduced to a bandform through a nite sequence of elementary uni-tary similarities. The two main results of this paper are the following assertions: (1) Any matrix with a low rank skew-Hermitian part is condens-able. (2) If A is a normal condensable matrix, then any rank one perturbation of A (generally nonnormal) is condensable as ...