Related papers: On Generalized Hilbert Matrices

In this paper we extend the orthogonal polynomials approach for extreme value calculations of Hermitian random matrices, developed by Nadal and Majumdar [1102.0738], to normal random matrices and 2D Coulomb gases in general. Firstly, we…

The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions -- the…

We use elementary triangular matrices to obtain some factorization, multiplication, and inversion properties of triangular matrices. We also obtain explicit expressions for the inverses of strict $k$-Hessenberg matrices and banded matrices.…

We propose a new iterative algorithm for generating a subset of eigenvalues and eigenvectors of large matrices which generalizes the method of optimal relaxations. We also give convergence criteria for the iterative process, investigate its…

For a semisimple, simply-connected linear algebraic group, $G$, and parabolic subgroup, $P\subseteq G$, we use the fact that the Hilbert polynomial of the equivariant embedding of $G/P$ is equal to the Hilbert function to compute an…

The inverse of a large matrix can often be accurately approximated by a polynomial of degree significantly lower than the order of the matrix. The iteration polynomial generated by a run of the GMRES algorithm is a good candidate, and its…

We study and derive algorithms for nonlinear eigenvalue problems, where the system matrix depends on the eigenvector, or several eigenvectors (or their corresponding invariant subspace). The algorithms are derived from an implicit…

This note presents absolute bounds on the size of the coefficients of the characteristic and minimal polynomials depending on the size of the coefficients of the associated matrix. Moreover, we present algorithms to compute more precise…

Hilbert order is widely applied in many areas. However, most of the algorithms are confined to low dimensional cases. In this paper, algorithms for encoding and decoding arbitrary dimensional Hilbert order are presented. Eight algorithms…

We study local generalized additive decompositions (GADs) of homogeneous polynomials and their associated point schemes through their local inverse systems. We prove that their construction and algebraic properties are independent of the…

We obtain an asymptotic formula for the number of integer $2\times 2$ matrices that have determinant $\Delta$ and whose absolute values of the entries are at most $H$. The result holds uniformly for a large range of $\Delta$ with respect to…

We propose a second-order accurate method to estimate the eigenvectors of extremely large matrices thereby addressing a problem of relevance to statisticians working in the analysis of very large datasets. More specifically, we show that…

In this note we translate the pictorial description of Gulotta's efficient inverse algorithm (arXiv:0807.3012) into matrix operations, so that it can be implemented on a computer. As an application we point out that this in combination with…

Motivated by [9] we study the existence of the inverse of infinite Hermitian moment matrices associated with measures with support on the complex plane. We relate this problem to the asymptotic behaviour of the smallest eigenvalues of…

This paper gives an explicit method for computing the resultant of any sparse unmixed bivariate system with given support. We construct square matrices whose determinant is exactly the resultant. The matrices constructed are of hybrid…

Every real hyperbolic form in three variables can be realized as the determinant of a linear net of Hermitian matrices containing a positive definite matrix. Such representations are an algebraic certificate for the hyperbolicity of the…

The determinants of $\{\pm 1\}$-matrices are calculated by via the oriented hypergraphic Laplacian and summing over an incidence generalization of vertex cycle-covers. These cycle-covers are signed and partitioned into families based on…

In this short note, we present a novel method for computing exact lower and upper bounds of eigenvalues of a symmetric tridiagonal interval matrix. Compared to the known methods, our approach is fast, simple to present and to implement, and…

We generalize entanglement detection with covariance matrices for an arbitrary set of observables. A generalized uncertainty relation is constructed using the covariance and commutation matrices, then a criterion is established by…

A recursive method is derived to calculate all eigenvalue correlation functions of a random hermitian matrix in the large size limit, and after smoothing of the short scale oscillations. The property that the two-point function is…