Echelon Form Research Articles

Elimination and Factorization

Summary If a matrix A has rank r, then its row echelon form (from elimination) contains the identity matrix in its first r independent columns. How do we intepret the matrix F that appears in the remaining columns of that echelon form? The matrix F multiplies those first r independent columns of A to give its n–r dependent columns. Then F reveals bases for the row space and the nullspace of the original matrix A. And F is the key to the column-row factorization A = CR .

Irreducible Cartesian tensor decomposition: A computational approach

Cartesian tensors are widely used in physics and chemistry, especially for the formulation of linear and nonlinear spectroscopies as well as for molecular response properties. In this work, we review the problem of irreducible Cartesian tensor (ICT) decomposition of a generic Cartesian tensor of rank n into its irreducible parts, each characterized by a specific symmetry. The matrix formulation of the ICT decomposition is structurally similar to the problem of rotational averaging using isotropic Cartesian tensors. Analogously to the latter, the ICT decomposition can be considered as a problem of selecting a set of permutations of n indices that provides a linearly independent set of mappings between Cartesian tensor subspaces. This selection can be performed using a simple computational approach based on the reduced row echelon form (rref) algorithm. This protocol has been implemented in a computer code used to re-derive the already known ICT decomposition for 2 ≤ n ≤ 4. Finally, for the first time, we performed the explicit ICT decomposition of a Cartesian tensor of rank n = 5.

An algorithm to recognize echelon subgroups of a free group

We provide an algorithm that, given a finite set of generators for a subgroup H of a finitely generated free group F, determines whether H is echelon or not and, in case of affirmative answer, also computes a basis with respect to which H is in echelon form. This gives an answer to a question of Rosenmann. We also prove, by means of a counterexample, that intersection of two echelon subgroups needs not to be echelon, answering another question of Rosenmann.

Gaussian elimination for flexible systems of linear inclusions

Flexible systems are linear systems of inclusions in which the elements of the coefficient matrix are external numbers in the sense of nonstandard analysis. External numbers represent real numbers with small, individual error terms. Using Gaussian elimination, a flexible system can be put into a row-echelon form with increasing error terms on the right-hand side. Then parameters are assigned to the error terms and the resulting system is solved by common methods of linear algebra. The solution set may have indeterminacy not only in terms of linear spaces, but also of modules. We determine maximal robustness for flexible systems.

Enhancing Security for Text Message Cryptography based on Basis Vectors

Cryptography is the practice and study of hiding information from all but those with the means or key to decode the message. Also the area of cryptography employs many different means of transforming normal data in to unreadable form. This paper’s aim of the study was how to keep the digital data secretly and sent it secretly through insecure channels based on basis vectors, that an activity builds around one of the techniques that illustrate an application of a set of basis vectors called matrix to cryptography The method involves two matrices of which one is used to encode the encoding matrix and the other one to decode the decoding matrix. The characters, in the original message or stream are assigned numerical values and the matrix must be row reduce echelon form for use in decoding. The proposed method is very simple in its principle and has great potential to be applied to other situations where the exchange of message is done confidentially.

Crystal-Structure Matches in Solid-Solid Phase Transitions.

The exploration of solid-solid phase transition suffers from the uncertainty of how atoms in two crystal structures match. We devised a theoretical framework to describe and classify crystal-structure matches (CSM). Such description fully exploits the translational and rotational symmetries and is independent of the choice of supercells. This is enabled by the use of the Hermite normal form, an analog of reduced echelon form for integer matrices. With its help, exhausting all CSMs is made possible, which goes beyond the conventional optimization schemes. In an example study of the martensitic transformation of steel, our enumeration algorithm finds many candidate CSMs with lower strains than known mechanisms. Two long-sought CSMs accounting for the most commonly observed Kurdjumov-Sachs orientation relationship and the Nishiyama-Wassermann orientation relationship are unveiled. Given the comprehensiveness and efficiency, our enumeration scheme provide a promising strategy for solid-solid phase transition mechanism research.

Fast parallel computation of reduced row echelon form to find the minimum distance of linear codes

Fast parallel computation of reduced row echelon form to find the minimum distance of linear codes

A hesitant fuzzy multiple-attribute decision-making method considering rank relation

A decision matrix is typically used to express hesitant information when solving multi-attribute decision problems in an uncertain environment. To further investigate the decision problem, this paper takes the property of matrix rank as the starting point, introduces it into the hesitant fuzzy theory, presents the concept of the rank of the hesitant fuzzy decision matrix and discusses the related properties, and then studies the hesitant fuzzy linear relation, obtaining the attribute reduction method based on the hesitant fuzzy linear relation and applying it to the multi-attribute decision making. It adds to the theoretical understanding of the hesitant fuzzy decision matrix. The aggregation operator first transforms the hesitant fuzzy information into a comprehensive decision matrix, and the row echelon transformation determines the rank of the matrix. Second, the hesitant fuzzy linear relationship is obtained using the rank property. A new hesitant fuzzy matrix is obtained after attribute reduction based on the hesitant fuzzy linear relationship, and the alternatives are sorted using the TOPSIS method. Finally, the effectiveness and superiority of the proposed method are demonstrated through a comparison of actual case analysis and existing methods, and the expected research purpose is met.

An Elementary Proof that Reduced Row Echelon form of a Matrix is Unique

"An Elementary Proof that Reduced Row Echelon form of a Matrix is Unique." The College Mathematics Journal, ahead-of-print(ahead-of-print), pp. 1–2

Efficient Construction of Involutory Linear Combinations of Anticommuting Pauli Generators for Large-Scale Iterative Qubit Coupled Cluster Calculations.

We present an efficient method for construction of a fully anticommutative set of Pauli generators (elements of the Pauli group) from a commutative set of operators that are composed exclusively from Pauli x̂ operators (purely X generators) and sorted by an associated numerical measure, such as absolute energy gradients. Our approach uses the Gauss-Jordan elimination applied to a binary matrix that encodes the set of X generators to bring it to the reduced row-echelon form, followed by the construction of an anticommutative system in a standard basis by means of a modified Jordan-Wigner transformation and returning to the original basis. The algorithm complexity is linear in the size of the X set and quadratic in the number of qubits. The resulting anticommutative sets are used to construct the qubit coupled cluster Ansatz with involutory linear combinations of anticommuting Paulis (QCC-ILCAP) proposed in J. Chem. Theory Comput. 2021, 17 (1), 66-78. We applied the iterative qubit coupled cluster method with the QCC-ILCAP Ansatz to calculations of ground-state potential energy curves for symmetric stretching of the water molecule (36 qubits) and dissociation of N2 (56 qubits).

Incremental Interval Assignment by Integer Linear Algebra with Improvements

Interval Assignment (IA) is the problem of selecting the number of mesh edges (intervals) for each curve for conforming quad and hex meshing. The intervals x is fundamentally integer-valued. Many other approaches perform numerical optimization then convert a floating-point solution into an integer solution, which is slow and error prone. We avoid such steps: we start integer, and stay integer. Incremental Interval Assignment (IIA) uses integer linear algebra (Hermite normal form) to find an initial solution to the meshing constraints, satisfying the integer matrix equation Ax=b. Solving for reduced row echelon form provides integer vectors spanning the nullspace of A. We add vectors from the nullspace to improve the initial solution, maintaining Ax=b. Heuristics find good integer linear combinations of nullspace vectors that provide strict improvement towards variable bounds or goals. IIA always produces an integer solution if one exists. In practice we usually achieve solutions close to the user goals, but there is no guarantee that the solution is optimal, nor even satisfies variable bounds, e.g. has positive intervals. We describe several algorithmic changes since first publication that tend to improve the final solution. The software is freely available.

Interpolation on the cubed sphere with spherical harmonics

We consider Lagrange interpolation with Spherical Harmonics of data located at the equiangular Cubed Sphere nodes. An approach based on a suitable Echelon Form of the associated Vandermonde matrix is carried out. As an outcome, a particular subspace of Spherical Harmonics is defined. This subspace possesses a high-frequency truncation, reminiscent of the rhomboidal truncation. Numerical results show the interest of this approach in various contexts. In particular, several examples of resolution of the Poisson equation on the sphere are displayed.

An Echelon Form of Weakly Infeasible Semidefinite Programs and Bad Projections of the psd Cone

A weakly infeasible semidefinite program (SDP) has no feasible solution, but it has approximate solutions whose constraint violation is arbitrarily small. These SDPs are ill-posed and numerically often unsolvable. They are also closely related to “bad” linear projections that map the cone of positive semidefinite matrices to a nonclosed set. We describe a simple echelon form of weakly infeasible SDPs with the following properties: (i) it is obtained by elementary row operations and congruence transformations, (ii) it makes weak infeasibility evident, and (iii) it permits us to construct any weakly infeasible SDP or bad linear projection by an elementary combinatorial algorithm. Based on our echelon form, we generate a library of computationally very difficult SDPs. Finally, we show that some SDPs in the literature are in our echelon form, for example, the SDP from the sum-of-squares relaxation of minimizing Motzkin’s famous polynomial.

Step-by-Step Decoding of Binary Quasi-Reversible BCH Codes

Binary quasi-reversible BCH codes whose defining set contains consecutive elements from negative to positive integers have received considerable attention in recent years due to their efficient decoding suitable for a wide range of applications. This paper combines the concept of the weight and quasi-reversible structures to introduce two subclasses of BCH codes: odd-like/even-like quasi-reversible BCH codes. The step-by-step decoding of these codes is developed as follows: First, the weight evaluation of a received polynomial is able to judge whether the number of errors is odd or even, which helps to simplify the decoding processes. Second, based on Chiò’s pivotal condensation process which can be easily implemented in a parallel computing architecture, the determinant calculation of the band matrix instead of Peterson’s matrix in column-echelon form is faster. Third, a newly proposed non-monic error-locator polynomial is sparser than the conventional ones. As a consequence, the theoretical analysis and experimental results validate potential benefits in requiring fewer finite field additions and multiplications used in the decoding of binary odd-like/even-like quasi-reversible BCH codes up to half the minimum distance when compared with the narrow-sense BCH codes with small error-correcting capability.

The footprint form of a matrix: Definition, properties, and an application

We propose an alternative to the well-known row echelon form of a matrix, focused on its “footprint”, that is, on the position of both the leading and trailing nonzero entries of each row. When a matrix is in footprint form, its leading entries are all in different positions (as in the echelon form) but, in addition, its trailing entries are also in different positions. This new form has several interesting properties. In particular, a matrix in footprint form has the smallest footprint among the matrices obtainable from it through elementary row operations, and supports an efficient way to compute the rank of any of its submatrices with a particular shape. This is critical for our application: scoring potential variable orders for the decision diagram encoding all solutions to a set of linear integer constraints, with the goal of finding an order that minimizes the decision diagram size. We then experimentally evaluate the effectiveness of this scoring, on several problem instances small enough to compare all possible orders.

A Gauche Perspective on Row Reduced Echelon Form and Its Uniqueness

Using a left-to-right “sweeping” algorithm, we define the Gauche basis for the column space of a matrix M. Interpreting the row reduced echelon form (RREF) of M by Gauche means gives a direct proof of its uniqueness. A corollary shows that the (right) null space of M determines its row equivalence class, unmasks a sanitized version of the assertion “if two systems are solution equivalent they are row equivalent,” and presents the null space as a distinguished graph. We conclude with pedagogical reflections.

Introduction to Applied Algebra: Book Review of Chapter 8-Linear Equations (System of Linear Equations)

This chapter review presents two ideas and techniques in solving Systems of Linear Equations in the most simple minded straightforward manner to enable the student as well as the instructor to follow it independently with very little guidance. The focus is on using simpler and easier approaches such as Determinants; and Elementary Row Operations to solve Systems of Linear Equations. We found the solution set of a few systems of linear equations by a successive ratio of the determinant of all the matrices formed from replacing each column of the coefficient matrix by the right hand side vector and the determinant of the coefficient matrix repeatedly giving the values of the variables in the system in the order in which they appeared. Similarly, we also used the three types of elementary row operations namely; Row Swap; Scalar Multiplication; and Row Sum to find the solution set of systems of linear equations through row echelon form to reduced row echelon form to find the solution set of some systems of linear equations. Technical forms of systems of linear equations were used to illustrate the two approaches in finding their solution sets. In each approach we started by finding the coefficient matrices from the systems of linear equations.

LU and CR Elimination

The reduced row echelon form ${\bf rref}(A)$ has traditionally been used for classroom examples: small matrices $A$ with integer entries and low rank $r$. This paper creates a column-row rank-revealing factorization ${A=CR}$, with the first $r$ independent columns of $A$ in $C$ and the $r$ nonzero rows of ${\bf rref}(A)$ in $R$. We want to reimagine the start of a linear algebra course by helping students to see the independent columns of $A$ and the rank and the column space. If $B$ contains the first $r$ independent rows of $A$, then those rows of ${A=CR}$ produce ${B=WR}$. The $r$ by $r$ matrix $W$ has full rank $r$, where $B$ meets $C$. Then the triple factorization ${A=CW^{-1}B}$ treats columns and rows of $A$ ($C$ and $B$) in the same way.

Education

The Education section in this issue presents two contributions. The first paper, “$LU$ and $CR$ Elimination,” is written by Gilbert Strang and Cleve Moler. This is an expository paper which is helpful to instructors and students interested in linear algebra and matrix manipulations. The focus of the paper is the column-row factorization for any matrix $A$ of rank $r$. The matrix is represented as $A=CR$, where the matrix $C$ contains the first $r$ independent columns of $A$, and the matrix $R$ contains the nonzero rows of the reduced row echelon form of $A$. The authors explain how to obtain the factorization and contrast it with $LU$, $QR$ factorizations, the single-value decomposition, and another factorization, $A= CW^{-1}B$, which they call magic. In section 6, the authors point to an important timely application pertaining to randomized algorithms for a low-rank approximation of large sparse matrices. The paper is clearly and nicely written. It includes a discussion about relevant MATLAB functions, examples, and several references to randomized linear algebra literature. The second paper is “Newton's Method in Mixed Precision,” presented by C. T. Kelley. The paper discusses the impact of precision in calculations on the rate of convergence of Newton's method for solving a system of nonlinear equations $F(x)=0.$ The vector-function $F$ is differentiable and its Jacobian $F' (x)$ is nonsingular. It is assumed that the step $s$ in each iteration is determined by solving the linear equation $F' (x)s= - F(x)$ via Gaussian elimination with column pivoting. The paper starts with the classical result on the local convergence of the method under standard assumptions and then passes onto analyzing the effect of errors arising in calculation of the function and in approximation of the Jacobian. The author's derivations predict no significant difference in the convergence of the nonlinear iteration between a double-precision analytic Jacobian with double precision in the linear solver and a forward-difference approximate Jacobian with a single precision in the linear solver. An interesting part of the paper deals with the question of estimating the effect of the backward error in the solver. The author observes that the worst case estimates are too pessimistic and rarely seen in practice. Invoking recent techniques in probabilistic rounding analysis, he derives more realistic results. The theoretical statements are supplemented by a numerical illustration in section 3. An additional example demonstrates how the theory breaks down if the Jacobian is singular at the solution.

Новые функции в MathPartner 2021

We introduce new features in the MathPartner service that have recently become available to users. We highlight the functions for calculating both arithmetic-geometric mean and geometric-harmonic mean. They allow calculating complete elliptic integrals of the first kind. They are useful for solving many physics problems, for example, one can calculate the period of a simple pendulum. Next, one can calculate the modified arithmetic-geometric mean proposed by Semjon Adlaj. Consequently, one can calculate the complete elliptic integrals of the second kind as well as the circumference of an ellipse. Furthermore, one can also calculate the Sylvester matrices of the first and the second kind. Thus, by means of a few strings, one can calculate the resultant of two polynomials as well as the discriminant of a binary form. Some new matrix functions are also added. So, today the list of matrix functions includes the transpose, adjugate, conjugate, inverse, generalized inverse, and pseudo inverse of a matrix, the matrix determinant, the kernel, the echelon form, the characteristic polynomial, the Bruhat decomposition, the triangular LDU decomposition, which is an exact block recursive LU decomposition, the QR block recursive decomposition, and the singular value decomposition. In addition, two block-recursive functions have been implemented for calculating the Cholesky decomposition of symmetric positive-definite matrices: one function for sparse matrices with the standard multiplication algorithm and another function for dense matrices with multiplication according to the Winograd–Strassen algorithm. The linear programming problems can be solved too. So, the MathPartner service has become better and handy. It is freely available at http://mathpar.ukma.edu.ua/ as well as at http://mathpar.com/