This note contains two remarks. The first remark concerns the extension of the well-known Cayley representation of rotation matrices by skew symmetric matrices to rotation matrices admitting −1 as an eigenvalue and then to all orthogonal matrices. We review a method due to Hermann Weyl and another method involving multiplication by a diagonal matrix whose entries are +1 or −1. The second remark...

We derive a formula for the backward error of a complex number λ when considered as an approximate eigenvalue of a Hermitian matrix pencil or polynomial with respect to Hermitian perturbations. The same are also obtained for approximate eigenvalues of matrix pencils and polynomials with related structures like skew-Hermitian, ∗-even and ∗-odd. Numerical experiments suggest that in many cases th...

The discrete Schrödinger equation with a quasiperiodic dichotomous potential specified by the Fibonacci sequence is known to have a singular continuous eigenvalue spectrum with all states being critically localized. This equation can be transformed into a quasiperiodic skew product dynamical system. In this iterative mapping which is entirely equivalent to the Schrödinger problem, critically lo...

We prove that the family of measured dynamical systems which can be realised as uniquely ergodic minimal homeomorphisms on a given manifold (of dimension at least two) is stable under measured extension. As a corollary, any ergodic system with an irrational eigenvalue is isomorphic to a uniquely ergodic minimal homeomorphism on the two-torus. The proof uses the following improvement of Weiss re...

The notion of diffusion via a stochastic process has been used as a means to approximate desired underlying probability distributions. An important task in such an application is to design a drift so as to accelerate the convergence. This problem can be cast as a special type of inverse eigenvalue problem where the inverse of a given covariance matrix is to be perturbed by skew-symmetric matric...

1. Page 2741, caption of Fig. 2: three occurrences of ‘‘commutator’’ instead of ‘‘commutor’’, i.e. the caption should read Fig. 2. Eigenvalue decomposition of the commutor for the CDFT: (a) sorted eigenvalues of the commutor T1 for N 1⁄4 128. (b) Selected symmetric and skew-symmetric eigenvectors of the commutor for k 1⁄4 0,1, 2, 3. 2. Page 2743, fourth sentence after Eq. (14): ‘‘commutator’’ i...

two digraphs of same order are said to be skew-equienergetic if theirskew energies are equal. one of the open problems proposed by li andlian was to construct non-cospectral skew-equienergetic digraphs onn vertices. recently this problem was solved by ramane et al. inthis paper, we give some new methods to construct new skew-equienergeticdigraphs.

In this paper the problem of locating eigenvalues of negative Krein signature is considered for operators of the form JL, where J is skew-symmetric with bounded inverse and L is self-adjoint. A finite-dimensional matrix, hereafter referred to as the Krein matrix, associated with the eigenvalue problem JLu = λu is constructed with the property that if the Krein matrix has a nontrivial kernel for...