Related papers: On normal and structured matrices under unitary st…
Structure-preserving algorithms and algorithms with uniform error bound have constituted two interesting classes of numerical methods. In this paper, we blend these two kinds of methods for solving nonlinear Hamiltonian systems with highly…
Miniversal deformations for pairs of skew-symmetric matrices under congruence are constructed. To be precise, for each such a pair $(A,B)$ we provide a normal form with a minimal number of independent parameters to which all pairs of…
We analyze an algorithm for computing a skew-Hermitian logarithm of a unitary matrix. This algorithm is very easy to implement using standard software and it works well even for unitary matrices with no spectral conditions assumed. Certain…
We derive the necessary and sufficient conditions for the simple eigenvalues of rational matrix functions with symmetry structure to have the same normwise condition number with respect to arbitrary and structure-preserving perturbations.…
In complete analogy with the classical situation (which is briefly reviewed) it is possible to define bi-Hamiltonian descriptions for Quantum systems. We also analyze compatible Hermitian structures in full analogy with compatible Poisson…
We complete all local spinor norm computations for quaternionic skew-hermitian forms over the field of rational numbers. Examples of class number computations are provided.
We introduce a canonical form for reduced bases of integral closures of discrete valuation rings, and we describe an algorithm for computing a basis in reduced normal form. This normal form has the same applications as the Hermite normal…
We derive computable formulas for the structured backward errors of a complex number $\lambda$ when considered as an approximate eigenvalue of rational matrix polynomials that carry a symmetry structure. We consider symmetric,…
The solution of problems in physics is often facilitated by a change of variables. In this work we present neural transformations to learn symmetries of Hamiltonian mechanical systems. Maintaining the Hamiltonian structure requires novel…
We prove that all injective maps on positive complex matrices which preserve order and shrink spectrum are implemented by unitary or antiunitary conjugations. We show by counterexamples that all assumptions are indispensable. The result…
We give a canonical form of m-by-2-by-2 spatial matrices for equivalence over any field.
We determine the structure of linear maps on complex (real) square matrices sending unitary (orthogonal) matrices to multiples of unitary (orthogonal) matrices. The result is used to determine the linear preservers of matrix pairs…
This short but self-contained survey presents a number of elegant matrix/operator inequalities for general convex or concave functions, obtained with a unitary orbit technique. Jensen, sub or super-additivity type inequalities are…
Our contribution is two-folded. First, starting from the known fact that every real skew-Hamiltonian matrix has a real Hamiltonian square root, we give a complete characterization of the square roots of a real skew-Hamiltonian matrix W.…
Matrices over the ring of formal power series are considered. Normal forms with respect to various sub-groups of the two-sided transformations are constructed. The construction is based on the special property of the action: it induces a…
We consider the computation of two normal forms for matrices over the univariate polynomials: the Popov form and the Hermite form. For matrices which are square and nonsingular, deterministic algorithms with satisfactory cost bounds are…
We discuss the dynamical quantum systems which turn out to be bi-unitary with respect to the same alternative Hermitian structures in a infinite-dimensional complex Hilbert space. We give a necessary and sufficient condition so that the…
We study star operations for Iwahori-Hecke algebras and invariant hermitian forms for finite dimensional modules over (graded) affine Hecke algebras with a view towards a unitarity algorithm.
Hamiltonian systems of ordinary and partial differential equations are fundamental mathematical models spanning virtually all physical scales. A critical property for the robustness and stability of computational methods in such systems is…
We use recent results on algorithms for Markov decision problems to show that a canonical form for a generalized P-matrix can be computed, in some important cases, by a strongly polynomial algorithm.