This note is devoted to the spectral analysis of rank one perturbations of unitary and self-adjoint operators. We study the following question: given two cyclic (i.e., having simple spectrum) operators A and B, when is A equivalent to B up to a rank one perturbation? More precisely, when does there exist a unitary operator U such that rank (UAU∗−B) = 1? As usual, we are looking for an answer in...

Array-oriented programming languages such as APL [1] and J [2] pay special attention to manipulating array structures: rank-one vectors (sequences of values), rank-two matrices (which can be seen as rectangular sequences of sequences), rank-three cuboids (sequences of sequences of sequences), rank-zero scalars, and so on. One appealing consequence of this unification is the prospect of rank pol...

In this article, we propose a method for solving the trust-region subproblem when a limited-memory symmetric rank-one matrix is used in place of the true Hessian matrix. The method takes advantage of two shape-changing norms to decompose the trust-region subproblem into two separate problems, one of which has a closed-form solution and the other one is easy to solve. Sufficient conditions for g...

The second-order partial differential wave Equation (Cauchy’s first equation of motion), derived from Newton’s force equilibrium, describes a standing field consisting two waves propagating in opposite directions, and is, therefore, “two-way equation”. Due to the second order differentials analytical solutions only exist few cases. “binomial factorization” linear two-way operator into first-ord...

We prove that any two unitary operators with simple singular spectrum which contains the whole circle are unitarily equivalent up to a rank one operator.