Given a pair of distinct eigenvalues (λ1, λ2) of an n×n quadratic matrix polynomial Q(λ) with nonsingular leading coefficient and their corresponding eigenvectors, we show how to transform Q(λ) into a quadratic of the form [ Qd(λ) 0 0 q(λ) ] having the same eigenvalues as Q(λ), with Qd(λ) an (n− 1)× (n− 1) quadratic matrix polynomial and q(λ) a scalar quadratic polynomial with roots λ1 and λ2. ...

We present a new algorithm for reducing an arbitrary unitary matrix U into a sequence of elementary operations (operations such as controlled-nots and qubit rotations). Such a sequence of operations can be used to manipulate an array of quantum bits (i.e., a quantum computer). Our algorithm applies recursively a mathematical technique called the CS Decomposition to build a binary tree of matric...

A method based on the elementary operations algorithm (EOA) is developed that reduces a system matrix describing a discrete linear repetitive process to a 2-D nonsingular Roesser form such that all the input-output properties, including the transfer-function matrix, are preserved. Some areas for possible future use/application of the developed results will also be briefly discussed.

A block Toeplitz algorithm is proposed to perform the J-spectral factorization of a para-Hermitian polynomial matrix. The input matrix can be singular or indefinite, and it can have zeros along the imaginary axis. The key assumption is that the finite zeros of the input polynomial matrix are given as input data. The algorithm is based on numerically reliable operations only, namely computation ...

The first thing taught in Math 340 is Gaussian Elimination, i.e. the process of transforming a matrix to reduced row echelon form by elementary row operations. Because this process has the effect of multiplying the matrix by an invertible matrix it has produces a new matrix for which the solution space of the corresponding linear system is unchanged. This is made precise by Theorem 2.4 below. T...

When a matrix is represented as a product of "elementary" matrices, the matrix, its transpose, its inverse and inverse transpose are readily available for vector multiplication. By an "elementary matrix" is meant one formed from the identity matrix by replacing one column ; thus an elementary matrix can be compactly recorded by the subscript of the altered column and the values of the elements ...

An efficient method for the computation to high relative accuracy of the LDU decomposition of an n × n row diagonally dominant M–matrix is presented, assuming that the off–diagonal entries and row sums are given. This method costs an additional O(n) elementary operations over the cost of Gaussian elimination, and leads to a lower triangular, column diagonally dominant matrix and an upper triang...

PREFACE 1022 CHAPTER I. The sources of errors in a computation 1.1. The sources of errors. (A) Approximations implied by the mathematical model. (B) Errors in observational data. (C) Finitistic approximations to transcendental and implicit mathematical formulations. (D) Errors of computing instruments in carrying out elementary operations: "Noise." Round off errors. "Analogy" and digital comput...

Block variants of the Jacobi-Davidson method for computing a few eigenpairs of a large sparse matrix are known to improve the robustness of the standard algorithm when it comes to computing multiple or clustered eigenvalues. In practice, however, they are typically avoided because the total number of matrix-vector operations increases. In this paper we present the implementation of a block Jaco...

We introduce our Transform Imager Technology and Architecture. This approach allows for Retina and higher-level bio-inspired computation in a programmable architecture that still possesses similar high-fill factor pixels of APS imagers. This imager is capable of programmable matrix operations on the image, where we can represent the image as either a full matrix or using block matrix operations...