23.1 Let A be a nonsingular square matrix and let A = QR and A∗A = U∗U be QR and cholesky factorizations, respectively, with the usual normalization rjj, ujj > 0. Is it true or false that R = U? Solution: It is true. Since A is nonsingular, it will have a unique QR factorization with rjj > 0. Then we have A∗A = R∗Q∗QR = R∗R because Q is a unitary matrix. On the other hand, it is easy to see tha...

In this thesis, computationally efficient detection algorithms for the minimum mean square error (MMSE) estimation with successive interference cancellation (SIC), or with generalized decision feedback equalizer (GDFE) are considered in spatial multiplexing multiple-input multipleoutput (MIMO) systems. The MMSE-SIC and the MMSE-GDFE architectures are known to achieve the ergodic capacity of MIM...

A Delaunay decomposition is a cell decomposition in Rd for which each cell is inscribed in a Euclidean ball which is empty of all other vertices. This article introduces a generalization of the Delaunay decomposition in which the Euclidean balls in the empty ball condition are replaced by other families of regions bounded by certain quadratic hypersurfaces. This generalized notion is adaptable ...

A new stochastic subspace identification algorithm is developed with the help of a stochastic realization on a finite interval. First, a finite-interval realization algorithm is rederived via “block-LDL decomposition” for a finite string of complete covariance sequence. Next, a stochastic subspace identification method is derived by adapting the finiteinterval realization algorithm to incomplet...

Two methods to decompose block matrices analogous to Singular Matrix Decomposition are proposed, one yielding the so called economy decomposition, and other yielding the full decomposition. This method is devised to avoid handling matrices bigger than the biggest blocks, so it is particularly appropriate when a limitation on the size of matrices exists. The method is tested on a document-term m...

We review recently developed methods to efficiently utilize the Cholesky decomposition technique in electronic structure calculations. The review starts with a brief introduction to the basics of the Cholesky decomposition technique. Subsequently, examples of applications of the technique to ab inito procedures are presented. The technique is demonstrated to be a special type of a resolution-of...

Nakajima and Tanaka showed that the algebraic eigenvalue problem occurring in the discrete ordinate and matrix operator methods can be reduced to finding eigenvalues and eigenvectors of the product of two symmetric matrices, one of which is positive definite. Here, we show that the Cholesky decomposition of this positive definite matrix can be used to convert the eigenvalue problem into one inv...

Using a functional inequality, the essential spectrum and eigenvalues are estimated for Laplace-type operators on Riemannian vector bundles. Consequently, explicit upper bounds are obtained for the dimension of the corresponding L 2-harmonic sections. In particular, some known results concerning Gromov's theorem and the L 2-Hodge decomposition are considerably improved. 1. Introduction. Recall ...

Usage of the Sherman-Morrison-Woodbury formula to update linear systems after low rank modifications of the system matrix is widespread in machine learning. However, it is well known that this formula can lead to serious instabilities in the presence of roundoff error. If the system matrix is symmetric positive definite, it is almost always possible to use a representation based on the Cholesky...