Square matrix

About: Square matrix is a research topic. Over the lifetime, 5000 publications have been published within this topic receiving 92428 citations.

A Formula for the Exponential of a Real Skew-Symmetric Matrix of Order 4

Abstract: In this short paper the formula of the exponential matrix e A when A is a kew-symmetric real matrix of order 4 is derived. The formula is a generalization of the well known Rodrigues formula for skew-symmetric matrices of order 3.

Discriminative visualization by limited rank matrix learning

Abstract: We propose an extension of the recently introduced Generalized Matrix Learning Vector Quantization (GMLVQ) algorithm. The original algorithm provides a discriminative distance measure of relevance factors, aided by adaptive square matrices, which can account for correlations between different features and their importance for the classification. We extend the scheme to matrices of limited rank corresponding to lowdimensional representations of the data. This allows to incorporate prior knowledge of the intrinsic dimension and to reduce the number of adaptive parameters efficiently. The case of twoor three-dimensional representations constitutes an efficient visualization method. The identification of a suitable projection is not treated as a preprocessing step but as an integral part of the supervised training. Machine Learning Reports,Research group on Computational Intelligence, http://www.uni-leipzig.de/compint Discriminative Visualization by Limited Rank Matrix Learning

URV decomposition based structured methods for palindromic and even eigenvalue problems

Abstract: In this work numerical methods for the solution of two classes of structured generalized eigenvalue problems, Ax = �Bx, are developed. Those classes are the palindromic (B = A T ) and the even (A = A T , B = −B T ) eigenvalue problems. The spectrum of these problems is not arbitrary, rather do eigenvalues occur in pairs. We will construct methods for palindromic and even eigenvalue problems that are of cubic complexity and that are guaranteed to produce eigenvalues that are paired to working precision. At the heart of both methods is a new URV-type matrix decomposition, that simultaneously transforms three matrices to skew triangular form, i.e., to a form that is triangular with respect to the Northeast-Southwest diagonal. The algorithm to compute this URV decomposition uses several other methods to reduce a single square matrix to skew triangular form: the skew QR factorization and the skew QRQ T decomposition. Moreover, a method to compute the singular value decomposition of a complex, skew symmetric matrix is presented and used.

Solvability, controllability and observability of singular systems

Abstract: The problems of solvability, controllability, and observability for the singular systemKx(t)=Ax(t)+Bu(t) are studied, whereK is a singular, square matrix andu(t) is a complex vector function sufficiently differentiable. The classical theories of matrix pencils are first related to the solvability of singular systems. Then, the concepts of reachability, controllability, and observability of regular systems are extended to singular systems. Finally, the set of reachable states is described. The proposed matrix conditions for testing the controllability and observability of singular systems are simple and always feasible.

On the full exploitation of symmetry in periodic (as well as molecular) self-consistent-field ab initio calculations

Abstract: Use of symmetry can dramatically reduce the computational cost (running time and memory allocation) of self-consistent-field ab initio calculations for molecular and crystalline systems. Crucial for running time is symmetry exploitation in the evaluation of one- and two-electron integrals, diagonalization of the Fock matrix at selected points in reciprocal space, reconstruction of the density matrix. As regards memory allocation, full square matrices (overlap, Fock, and density) in the Atomic Orbital (AO) basis are avoided and a direct transformation from the packed AO to the symmetry adapted crystalline orbital basis is performed, so that the largest matrix to be handled has the size of the largest sub-block in the latter basis. Quantitative examples, referring to the implementation in the CRYSTAL code, are given for high symmetry families of compounds such as carbon fullerenes and nanotubes.