# Registered Data

Contents

- 1 [CT008]
- 1.1 [02200] Structured Distances to Nearest Singular Matrix Pencil
- 1.2 [00430] Nearest singular pencil via Riemannian optimization
- 1.3 [01156] Row completion of polynomial matrices
- 1.4 [01208] On bundles of matrix pencils under strict equivalence
- 1.5 [00770] Rellich eigendecomposition of paraHermitian matrices, with applications

# [CT008]

## [02200] Structured Distances to Nearest Singular Matrix Pencil

**Session Date & Time**: 1E (Aug.21, 17:40-19:20)**Type**: Contributed Talk**Abstract**: We consider the structured distance to singularity for a given regular matrix pencil $A+sE$, where $(A,E)\in \mathbb S \subseteq (\mathbb{C}^{n,n})^2$. This includes Hermitian, skew-Hermitian, $*$-even, $*$-odd, $*$-palindromic, T-palindromic, and dissipative Hamiltonian pencils. We derive explicit computable formulas for the distance to the nearest structured pencil $(A-\Delta_A)+s(E-\Delta_E)$ such that $A-\Delta_A$ and $E-\Delta_E$ have a common null vector. We then obtain a family of computable lower bounds for the unstructured and structured distances to singularity.**Classification**:__15A18__,__15A22__,__65K05__**Author(s)**:**Anshul Prajapati**(Indian Institute of Technology Delhi)- Punit Sharma (Indian Institute of Technology Delhi)

## [00430] Nearest singular pencil via Riemannian optimization

**Session Date & Time**: 1E (Aug.21, 17:40-19:20)**Type**: Contributed Talk**Abstract**: The problem of finding the nearest complex $($real$)$ singular pencil can be cast as a minimization problem over the manifold $U(n) \times U(n)$ $\left( O(n) \times O(n) \right)$ via the generalized Schur form. This novel perspective yields a competitive numerical method by pairing it with an algorithm capable of doing optimization on a Riemannian manifold.**Classification**:__15A22__**Author(s)**:**Lauri Nyman**(Aalto University)

## [01156] Row completion of polynomial matrices

**Session Date & Time**: 1E (Aug.21, 17:40-19:20)**Type**: Contributed Talk**Abstract**: Perturbation problems arise frequently in applications, as in structural changes of the dynamics of a system or in pole placement problems in control theory. Perturbation problems of matrices are closely related to completion problems. We present a solution to the row-completion problem of a polynomial matrix, prescribing the eigenstructure of the resulting matrix and maintaining the degree.**Classification**:__15A22__,__15A83__**Author(s)**:- Agustzane Amparan (Universidad del País Vasco, UPV/EHU)
- Itziar Baragaña (Universidad del País Vasco, UPV/EHU)
- Silvia Marcaida (Universidad del País Vasco, UPV/EHU)
**Alicia Roca**(Universitat Politècnica de València )

## [01208] On bundles of matrix pencils under strict equivalence

**Session Date & Time**: 1E (Aug.21, 17:40-19:20)**Type**: Contributed Talk**Abstract**: Bundles of matrix pencils under strict equivalence are sets of pencils having the same Kronecker canonical form, up to the eigenvalues: namely, they are unions of orbits. The notion of bundle was introduced by Arnold in 1971, and it has been extensively used since then. In this talk, we review and/or formalize some notions and results on bundles of pencils and provide a characterization for the inclusion relation between their closures in the standard topology.**Classification**:__15A22__,__15A18__,__15A21__,__15A54__,__65F15__**Author(s)**:**FERNANDO DE TERÁN**(Universidad Carlos III de Madrid)- FROILÁN M. DOPICO (Universidad Carlos III de Madrid)

## [00770] Rellich eigendecomposition of paraHermitian matrices, with applications

**Session Date & Time**: 1E (Aug.21, 17:40-19:20)**Type**: Contributed Talk**Abstract**: Let $H(z)$ be paraHermitian, that is, analytic and Hermitian on the unit circle $S^1$. We prove that $H(z)=U(z)D(z)U(z)^P$ where, for all $z \in S^1$, $U(z)$ is unitary, $U(z)^P=U(z)^*$, and $D(z)$ is real diagonal; moreover, $U(z), D(z)$ are analytic in $w=z^{1/N}$ for some positive integer $N$, and $U(z)^P$ is the paraHermitian conjugate of $U(z)$. We discuss the implications on the svd of an $S^1$-analytic matrix and the sign characteristics of unimodular eigenvalues of $*$-palindromic matrix polynomials.**Classification**:__15A23__,__15A18__,__15A54__,__15B57__**Author(s)**:**Vanni Noferini**(Aalto University)- Giovanni Barbarino (Aalto University)