scholarly journals Some classes of matrix transforms of summability domains of normal matrices

Filomat ◽  
2012 ◽  
Vol 26 (5) ◽  
pp. 1023-1028 ◽  
Author(s):  
Ants Aasm
Keyword(s):  
Triangular Matrix ◽  
Normal Matrix ◽  
Regular Matrix ◽  
Normal Matrices ◽  
Classes Of Matrices ◽  
Class Of Matrices ◽  
Matrix Transforms ◽  
Special Case

Let A, B and M be matrices with real or complex entries. In this paper two classes of matrices M, transforming the summability domain of A into the summability domain of B, are characterized. The M-consistency of A and B on the summability domain of A for a triangular matrix A and a regular matrix B is considered. As an application of the main results of this paper one class of matrices M, transforming the summability domain of A into the summability domain of B in the special case if B is the Ces?ro matrix and A is a normal matrix, is described.

Quasi-normal matrices and products

1970 ◽  
Vol 11 (3) ◽  
pp. 329-339 ◽  
Author(s):  
N. A. Wiegmann
Keyword(s):  
Canonical Form ◽  
Normal Matrix ◽  
Complex Conjugate ◽  
Complex Matrix ◽  
Normal Matrices ◽  
Basic Properties ◽  
The Matrix ◽  
Conjugate Transpose ◽  
Class Of Matrices

A normal matrix A = (aij) with complex elements is a matrix such that AACT = ACTA where ACT denotes the (complex) conjugate transpose of A. In an article by K. Morita [2] a quasi-normal matrix is defined to be a complex matrix A which is such that AACT = ATAC, where T denotes the transpose of A and AC the matrix in which each element is replaced by its conjugate, and certain basic properties of such a matrix are developed there. (Some doubt might exist concerning the use of ‘quasi’ since this class of matrices does not contain normal matrices as a sub-class; however, in deference to the original paper and the normal canonical form of Theorem 1 below, the terminology in [2] is used.)

Summability of Matrix Transforms of Subsequences

1981 ◽  
Vol 24 (3) ◽  
pp. 359-364
Author(s):  
Thomas A. Keagy
Keyword(s):  
Limit Point ◽  
Regularity Condition ◽  
Summability Method ◽  
Finite Limit ◽  
Regular Matrix ◽  
Matrix Summability ◽  
Matrix Transforms

AbstractD. F. Dawson has considered several questions of the following nature. Suppose T is a regular matrix summability method. If A is a regular matrix and x is a sequence having a finite limit point, then there exists a subsequence y of x such that each finite limit point of x is a T-limit point of Ay. In the present paper, we show the regularity condition for A may be replaced by the requirement that A be a limit preserving bv to c map. This leads to summability characterizations for several classes of sequences.

Nonconvexity of the Generalized Numerical Range Associated with the Principal Character

2000 ◽  
Vol 43 (4) ◽  
pp. 448-458
Author(s):  
Chi-Kwong Li ◽  
Alexandru Zaharia
Keyword(s):  
Symmetric Group ◽  
Matrix Function ◽  
Numerical Range ◽  
Complex Matrix ◽  
Normal Matrices ◽  
Principal Character ◽  
Straight Line ◽  
Generalized Matrix ◽  
Generalized Numerical Range ◽  
Special Case

AbstractSuppose m and n are integers such that 1 ≤ m ≤ n. For a subgroup H of the symmetric group Sm of degree m, consider the generalized matrix function on m × m matrices B = (bij) defined by and the generalized numerical range of an n × n complex matrix A associated with dH defined byIt is known that WH(A) is convex if m = 1 or if m = n = 2. We show that there exist normal matrices A for which WH(A) is not convex if 3 ≤ m ≤ n. Moreover, for m = 2 < n, we prove that a normal matrix A with eigenvalues lying on a straight line has convex WH(A) if and only if νA is Hermitian for some nonzero ν ∈ ℂ. These results extend those of Hu, Hurley and Tam, who studied the special case when 2 ≤ m ≤ 3 ≤ n and H = Sm.

scholarly journals Unique builders for classes of matrices

2021 ◽  
Vol 9 (1) ◽  
pp. 52-65
Author(s):  
Ted Hurley
Keyword(s):  
Quantum Logic ◽  
Logic Gates ◽  
Building Blocks ◽  
Unitary Matrix ◽  
Hermitian Matrices ◽  
Unique Form ◽  
Quantum Logic Gates ◽  
Normal Matrices ◽  
Classes Of Matrices ◽  
Pseudo Inverse

Abstract Basic matrices are defined which provide unique building blocks for the class of normal matrices which include the classes of unitary and Hermitian matrices. Unique builders for quantum logic gates are hence derived as a quantum logic gates is represented by, or is said to be, a unitary matrix. An efficient algorithm for expressing an idempotent as a unique sum of rank 1 idempotents with increasing initial zeros is derived. This is used to derive a unique form for mixed matrices. A number of (further) applications are given: for example (i) U is a symmetric unitary matrix if and only if it has the form I − 2E for a symmetric idempotent E, (ii) a formula for the pseudo inverse in terms of basic matrices is derived. Examples for various uses are readily available.

scholarly journals EFFICIENT COMPUTATION OF POSTERIOR COVARIANCE IN BUNDLE ADJUSTMENT IN DBAT FOR PROJECTS WITH LARGE NUMBER OF OBJECT POINTS

2020 ◽  
Vol XLIII-B2-2020 ◽  
pp. 737-744
Author(s):  
N. Börlin ◽  
A. Murtiyoso ◽  
P. Grussenmeyer
Keyword(s):  
Computational Effort ◽  
Bundle Adjustment ◽  
Normal Matrix ◽  
Normal Equation ◽  
Data Sets ◽  
Efficient Computation ◽  
Data Set ◽  
Normal Matrices ◽  
Self Calibration ◽  
Significant Addition

Abstract. One of the major quality control parameters in bundle adjustment are the posterior estimates of the covariance of the estimated parameters. Posterior covariance computations have been part of the open source Damped Bundle Adjustment Toolbox in Matlab (DBAT) since its first public release. However, for large projects, the computation of especially the posterior covariances of object points have been time consuming.The non-zero structure of the normal matrix depends on the ordering of the parameters to be estimated. For some algorithms, the ordering of the parameters highly affect the computational effort needed to compute the results. If the parameters are ordered to have the object points first, the non-zero structure of the normal matrix forms an arrowhead.In this paper, the legacy DBAT posterior computation algorithm was compared to three other algorithms: The Classic algorithm based on the reduced normal equation, the Sparse Inverse algorithm by Takahashi, and the novel Inverse Cholesky algorithm. The Inverse Cholesky algorithm computes the explicit inverse of the Cholesky factor of the normal matrix in arrowhead ordering.The algorithms were applied to normal matrices of ten data sets of different types and sizes. The project sizes ranged from 21 images and 100 object points to over 900 images and 400,000 object points. Both self-calibration and non-self-calibration cases were investigated. The results suggest that the Inverse Cholesky algorithm is the fastest for projects up to about 300 images. For larger projects, the Classic algorithm is faster. Compared to the legacy DBAT implementation, the Inverse Cholesky algorithm provides a performance increase by one to two orders of magnitude. The largest data set was processed in about three minutes on a five year old workstation.The legacy and Inverse Cholesky algorithms were implemented in Matlab. The Classic and Sparse Inverse algorithms included code written in C. For a general toolbox as DBAT, a pure Matlab implementation is advantageous, as it removes any dependencies on, e.g., compilers. However, for a specific lab with mostly large projects, compiling and using the classic algorithm will most likely give the best performance. Nevertheless, the Inverse Cholesky algorithm is a significant addition to DBAT as it enables a relatively rapid computation of more statistical metrics, further reinforcing its application for reprocessing bundle adjustment results of black-box solutions.

scholarly journals MPI+OpenMP implementation of the BiCGStab method with explicit preconditioning for the numerical solution of sparse linear systems

2019 ◽  
pp. 516-527
Author(s):  
И.Е. Капорин ◽  
О.Ю. Милюкова
Keyword(s):  
Parallel Implementation ◽  
Sparse Matrix ◽  
Triangular Matrix ◽  
Positive Definite Matrix ◽  
Test Problems ◽  
Approximate Inverse ◽  
University Of Florida ◽  
Upper Triangular Matrix ◽  
Class Of Matrices ◽  
The University

Для предобусловливания несимметричной положительно определенной разреженной матрицы рассматривается ее приближенная обратная, представленная в виде произведения нижнетреугольной и верхнетреугольной матриц. Предлагается новый способ предобусловливания положительно определенной разреженной матрицы--- метод блочного Якоби неполного обратного LU-разложения. Описан алгоритм параллельной реализации метода BiCGStab с предложенным предобусловливанием с применением MPIOpenMP-технологии. Проводится сравнение времени решения тестовых задач из коллекции разреженных матриц SuiteSparse (ранее известной как коллекция университета Флориды) методом BiCGStab с предложенным предобусловливанием и с предобусловливанием Якоби, а также с предобусловливанием блочного Якоби в сочетании с неполным треугольным разложением без заполнения. При этом используются разработанные параллельные реализации на основе MPI- или MPIOpenMP-подходов. A preconditioner for a large sparse nonsymmetric positive definite matrix is considered on the basis of its approximate inverse in the form of product of a lower triangular sparse matrix by an upper triangular matrix. For the class of matrices being considered, a new preconditioning based on the approximate block Jacobi with incomplete inverse LU-factorization preconditioning is proposed. For a parallel implementation of the corresponding preconditioned BiCGStab algorithm, the MPIOpenMP techniques are used. The timing results obtained for the MPIOpenMP and MPI implementations of the proposed preconditioning and for the Jacobi preconditioning used with the BiCGStab are compared using several test problems from the SuiteSparse collection (formerly known as the University of Florida sparse matrix collection).

A Hardy-Davies-Petersen Inequality for a Class of Matrices

1978 ◽  
Vol 30 (03) ◽  
pp. 458-465 ◽  
Author(s):  
P. D. Johnson ◽  
R. N. Mohapatra
Keyword(s):  
Triangular Matrix ◽  
Diagonal Matrix ◽  
Lower Triangular Matrix ◽  
Lower Triangular ◽  
Class Of Matrices

Let ω be the set of all real sequences a = ﹛an﹜ n ≧0. Unless otherwise indicated operations on sequences will be coordinatewise. If any component of a has the entry oo the corresponding component of a-1 has entry zero. The convolution of two sequences s and q is given by s * q . The Toeplitz martix associated with sequence s is the lower triangular matrix defined by tnk = sn-k (n ≧ k), tnk = 0 (n &lt; k). It can be seen that Ts(q) = s * q for each sequence q and that Ts is invertible if and only if s0 ≠ 0. We shall denote a diagonal matrix with diagonal sequence s by Ds.

scholarly journals On optimal symmetric orthogonalisation and square roots of a normal matrix

1993 ◽  
Vol 47 (2) ◽  
pp. 233-246
Author(s):  
Nagwa Sherif
Keyword(s):  
Best Approximation ◽  
Polar Decomposition ◽  
Full Rank ◽  
Normal Matrix ◽  
Iterative Technique ◽  
Approximation Properties ◽  
Square Roots ◽  
Normal Matrices

It is well known that the factors in the polar decomposition of a full rank real m × n matrix, m ≥ n possess best approximation properties. We propose an iterative technique to compute the polar factors based on these best approximation properties. For normal matrices, the polar decomposition is useful. It is applied to compute the principal square roots of real and complex normal matrices.

scholarly journals Positive linear maps on normal matrices

2018 ◽  
Vol 29 (12) ◽  
pp. 1850088 ◽  
Author(s):  
Jean-Christophe Bourin ◽  
Eun-Young Lee
Keyword(s):  
Normal Matrix ◽  
Linear Combinations ◽  
Linear Map ◽  
Normal Matrices ◽  
Schur Product ◽  
Linear Maps ◽  
Positive Linear Map ◽  
Matrix Formula ◽  
Positive Linear Maps ◽  
Unitary Orbit

For a positive linear map [Formula: see text] and a normal matrix [Formula: see text], we show that [Formula: see text] is bounded by some simple linear combinations in the unitary orbit of [Formula: see text]. Several elegant sharp inequalities are derived, for instance for the Schur product of two normal matrices [Formula: see text], [Formula: see text] for some unitary [Formula: see text], where the constant [Formula: see text] is optimal.

scholarly journals Summability of subsequences of a divergent sequence by regular matrices II

Filomat ◽  
2019 ◽  
Vol 33 (14) ◽  
pp. 4509-4517
Author(s):  
Johann Boos
Keyword(s):  
Positive Density ◽  
Regular Matrix ◽  
Divergent Sequence ◽  
Second Category ◽  
Special Case ◽  
Divergent Sequences

C. Stuart proved in [27, Proposition 7] that the Ces?ro matrix C1 cannot sum almost every subsequence of a bounded divergent sequence. At the end of the paper he remarked ?It seems likely that this proposition could be generalized for any regular matrix, but we do not have a proof of this?. In [4, Theorem 3.1] Stuart?s conjecture is confirmed, and it is even extended to the more general case of divergent sequences. In this note we show that [4, Theorem 3.1] is a special case of Theorem 3.5.5 in [24] by proving that the set of all index sequences with positive density is of the second category. For the proof of that a decisive hint was given to the author by Harry I. Miller a few months before he passed away on 17 December 2018.

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