The article ”Random matrix theory in statistics: A review” was written by D. Paul and A. Aue and published in the Journal of Statistical Planning and Inference in 2015. Random Matrix Theory (RMT) is interested among other topics in describing the asymptotic behavior of the singular values and singular vectors of random matrices. Random matrices emerge in many statistical problems, that can be t...

We consider discretizations of the hyper-singular integral operator on closed surfaces and show that the inverses of the corresponding system matrices can be approximated by blockwise low-rank matrices at an exponential rate in the block rank. We cover in particular the data-space format of H-matrices. We show the approximability result for two types of discretizations. The first one is a saddl...

One of the aims of this paper is to solve an open problem of Lovász about relations between graph spectra and cut-distance. The paper starts with several inequalities between two versions of the cut-norm and the two largest singular values of arbitrary complex matrices, extending, in particular, the well-known graph-theoretical Expander Mixing Lemma and giving a hitherto unknown converse of it....

The computation of solution paths for continuation problems requires the solution of a sequence of nonlinear systems of equations. Each nonlinear system can be solved by computing the solution of a succession of linear systems of equations determined by Jacobian matrices associated with the nonlinear system of equations. Points on the solution path where the Jacobian matrix is singular are refe...

In this paper, we present FPT-algorithms for special cases of the shortest vector problem (SVP) and the integer linear programming problem (ILP), when matrices included to the problems’ formulations are near square. The main parameter is the maximal absolute value of rank minors of matrices included to the problem formulation. Additionally, we present FPT-algorithms with respect to the same mai...

We present two generalizations of Singular Value Decomposition from real-numbered matrices to dual-numbered matrices. prove that every matrix has both types SVD. Both our are motivated by applications, either geometry or mechanics.

A procedure to obtain differentiation matrices with application to solve boundary value problems and to find limit-cycles of nonautonomous dynamical systems is extended straightforwardly to yield new differentiation matrices useful to obtain derivatives of complex rational functions. Such matrices can be used to obtain numerical solutions of some singular differential problems defined in the co...

In this paper, we present FPT-algorithms for special cases of the shortest lattice vector, integer linear programming, and simplex width computation problems, when matrices included in the problems' formulations are near square. The parameter is the maximum absolute value of rank minors of the corresponding matrices. Additionally, we present FPT-algorithms with respect to the same parameter for...

In this paper we show new formulas for the spectral radius and the spectral subradius of a set of matrices. The advantage of our results is that we express the spectral radius of any set of matrices by the spectral radius of a set of symmetric positive definite matrices. In particular, in one of our formulas the spectral radius is expressed by singular eigenvalues of matrices, whereas in the ex...

The aim of this paper is to explore to which extend the theory of the Hamburger moment problem for real Jacobi matrices generalizes to the case of complex Jacobi matrices. In particular, we characterize the indeterminacy in terms of uniqueness of closed extensions of Jacobi matrices, and discuss the link to the growth of the smallest singular values of the underlying Hankel matrices. As a bypro...