Till Örebro universitet

Motion capture (MoCap) technology is an essential tool for recording and analyzing movements of objects or humans. However, MoCap systems frequently encounter the challenge of missing data, stemming from mismatched markers, occlusion, or equipment limitations. Recovery of these missing data is imperative to maintain the reliability and integrity of MoCap recordings. This paper introduces a novel application of the tensor framework for MoCap data completion. We propose three completion algorithms based on the canonical polyadic (CP) decomposition of tensors. The first algorithm utilizes CP decomposition to capture the low-rank structure of the tensor. However, relying only on low-rank assumptions may be insufficient to deal with complex motion data. Thus, we propose two modified CP decompositions that incorporate additional information, SmoothCP and SparseCP decompositions. SmoothCP integrates piecewise smoothness prior, while SparseCP incorporates sparsity prior, each aiming to improve the accuracy and robustness of MoCap data recovery. To compare and evaluate the merit of the proposed algorithms over other tensor completion methods in terms of several evaluation metrics, we conduct numerical experiments with different MoCap sequences from the CMU motion capture dataset.

We show that the set of m x m complex skew-symmetric matrix polynomials of even grade d, i.e., of degree at most d, and (normal) rank at most 2r is the closure of the single set of matrix polynomials with certain, explicitly described, complete eigenstructure. This complete eigenstructure corresponds to the most generic m x m complex skew-symmetric matrix polynomials of even grade d and rank at most 2r. The analogous problem for the case of skew-symmetric matrix polynomials of odd grade is solved in [24].

We obtain the generic complete eigenstructures of complex Hermitian n x n matrix pencils with rank at most r (with r <= n). To do this, we prove that the set of such pencils is the union of a finite number of bundle closures, where each bundle is the set of complex Hermitian n x n pencils with the same complete eigenstructure (up to the specific values of the distinct finite eigenvalues). We also obtain the explicit number of such bundles and their codimension. The cases r = n, corresponding to general Hermitian pencils, and r < n exhibit surprising differences, since for r < n the generic complete eigenstructures can contain only real eigenvalues, while for r = n they can contain real and nonreal eigenvalues. Moreover, we will see that the sign characteristic of the real eigenvalues plays a relevant role for determining the generic eigenstructures.

In this paper, we consider the solvability conditions of some Sylvester-type quaternion matrix equations. We establish some practical necessary and sufficient conditions for the existence of solutions of a Sylvester-type quaternion matrix equation with five unknowns through the corresponding equivalence relations of the block matrices. Moreover, we present some solvability conditions to some Sylvester-type quaternion matrix equations, including those involving Hermicity. The findings of this article extend related known results.

Schur decompositions and the corresponding Schur forms of a single matrix, a pair of matrices, or a collection of matrices associated with the periodic eigenvalue problem are frequently used and studied. These forms are upper-triangular complex matrices or quasi-upper-triangular real matrices that are equivalent to the original matrices via unitary or, respectively, orthogonal transformations. In general, for theoretical and numerical purposes we often need to reduce, by admissible transformations, a collection of matrices to the Schur form. Unfortunately, such a reduction is not always possible. In this paper we describe all collections of complex (real) matrices that can be reduced to the Schur form by the corresponding unitary (orthogonal) transformations and explain how such a reduction can be done. We prove that this class consists of the collections of matrices associated with pseudoforest graphs. In other words, we describe when the Schur form of a collection of matrices exists and how to find it.

The Gaussian phase-space representation can be used to implement quantum dynamics for fermionic particles numerically. To improve numerical results, we explore the use of dynamical diffusion gauges in such implementations. This is achieved by benchmarking quantum dynamics of few-body systems against independent exact solutions. A diffusion gauge is implemented here as a so-called noise-matrix, which satisfies a matrix equation defined by the corresponding Fokker-Planck equation of the phase-space representation. For the physical systems with fermionic particles considered here, the numerical evaluation of the new diffusion gauges allows us to double the practical simulation time, compared with hitherto known analytic noise-matrices. This development may have far reaching consequences for future quantum dynamical simulations of many-body systems. 

A number of theoretical and computational problems for matrix polynomials are solved by passing to linearizations. Therefore a perturbation theory, that relates perturbations in the linearization to equivalent perturbations in the corresponding matrix polynomial, is needed. In this paper we develop an algorithm that finds which perturbation of matrix coefficients of a matrix polynomial corresponds to a given perturbation of the entire linearization pencil. Moreover we find transformation matrices that, via strict equivalence, transform a perturbation of the linearization to the linearization of a perturbed polynomial. For simplicity, we present the results for the first companion linearization but they can be generalized to a broader class of linearizations.

Recent years have seen a renewal of interest in multi-term linear matrix equations, as these have come to play a role in a number of important applications. Here, we consider the solution of such equations by means of the dynamical functional particle method, an iterative technique that relies on the numerical integration of a damped second order dy-namical system. We develop a new algorithm for the solution of a large class of these equations, a class that includes, among others, all linear matrix equations with Hermi-tian positive definite or negative definite coefficients. In numerical experiments, our MAT -LAB implementation outperforms existing methods for the solution of multi-term Sylvester equations. For the Sylvester equation AX + XB = C, in particular, it can be faster and more accurate than the built-in implementation of the Bartels-Stewart algorithm, when A and B are well conditioned and have very different size.

We show that the set of n x n complex symmetric matrix pencils of rank at most r is the union of the closures of left perpendicular r/2 Right perpendicular + 1 sets of matrix pencils with some, explicitly described, complete eigenstructures. As a consequence, these are the generic complete eigenstructures of n x n complex symmetric matrix pencils of rank at most r. We also show that the irreducible components of the set of n x n symmetric matrix pencils with rank at most r, when considered as an algebraic set, are among these closures.

We determine the generic complete eigenstructures for n x n complex symmetric matrix polynomials of odd grade d and rank at most r. More precisely, we show that the set of n \times n complex symmetric matrix polynomials of odd grade d, i.e., of degree at most d, and rank at most r is the union of the closures of the left perpendicularrd/2right perpendicular + 1 sets of symmetric matrix polynomials having certain, explicitly described, complete eigenstructures. Then we prove that these sets are open in the set of n x n complex symmetric matrix polynomials of odd grade d and rank at most r. In order to prove the previous results, we need to derive necessary and sufficient conditions for the existence of symmetric matrix polynomials with prescribed grade, rank, and complete eigenstructure in the case where all their elementary divisors are different from each other and of degree 1. An important remark on the results of this paper is that the generic eigenstructures identified in this work are completely different from the ones identified in previous works for unstructured and skew-symmetric matrix polynomials with bounded rank and fixed grade larger than 1, because the symmetric ones include eigenvalues while the others not. This difference requires using new techniques.