Linear Algebra Concepts Research Articles

Eigensolution of Laplacian matrices for graph partitioning and domain decomposition

PurposeThe purpose of this paper is to introduce a general equation for eigensolution. Eigenvalues and eigenvectors of graphs have many applications in combinatorial optimization and structural mechanics. Some important applications of graph products consist of nodal ordering and graph partitioning for structuring the structural matrices and finite element subdomaining, respectively.Design/methodology/approachIn the existing methods for the eigensolution of Laplacian matrices, members have been added to the model of a graph product such that for its Laplacian matrix an algebraic relation between blocks become possible. These methods are categorized as topological approaches. Here, using concepts of linear algebra a general algebraic method is developed.FindingsA new algebraic method is introduced for calculating the eigenvalues of Laplacian matrices in graph products.Originality/valueThe present method provides a simple tool for calculating the eigenvalues of the Laplacian matrices without using the configurational model and merely by using the Laplacian matrices. The developed formula for calculating the eigenvalues contains approximate terms which can be managed by the analyst.

Use of models in the teaching of linear algebra

We report results on an approach to teaching linear algebra using models. In particular we are interested in analyzing the use of two theories of mathematics education, namely, Models and Modeling and APOS in the design of a teaching sequence that starts with the proposal of a “real life” decision making problem to the students. We briefly illustrate the possibilities of this methodology through the analysis and description of our classroom experience on a problem related to traffic flow that elicits the use of a system of linear equations and different parameterizations of this system to answer questions on traffic control. We describe cycles of students’ work on the problem and discuss the advantages of this approach in terms of students’ learning and the possibilities of extending it to other problems and linear algebra concepts.

Jordan Canonical Form: Theory and Practice

Jordan Canonical Form (JCF) is one of the most important, and useful, concepts in linear algebra. The JCF of a linear transformation, or of a matrix, encodes all of the structural information about that linear transformation, or matrix. This book is a careful development of JCF. After beginning with background material, we introduce Jordan Canonical Form and related notions: eigenvalues, (generalized) eigenvectors, and the characteristic and minimum polynomials. We decide the question of diagonalizability, and prove the Cayley-Hamilton theorem. Then we present a careful and complete proof of the fundamental theorem: Let V be a finite-dimensional vector space over the field of complex numbers C, and let T : V → V be a linear transformation. Then T has a Jordan Canonical Form. This theorem has an equivalent statement in terms of matrices: Let A be a square matrix with complex entries. Then A is similar to a matrix J in Jordan Canonical Form, i.e., there is an invertible matrix P and a matrix J in Jordan Canonical Form with A = PJP-1. We further present an algorithm to find P and J, assuming that one can factor the characteristic polynomial of A. In developing this algorithm we introduce the eigenstructure picture (ESP) of a matrix, a pictorial representation that makes JCF clear. The ESP of A determines J, and a refinement, the labeled eigenstructure picture (lESP) of A, determines P as well. We illustrate this algorithm with copious examples, and provide numerous exercises for the reader.

A spectral proof of the uniqueness of a strongly regular graph with parameters [formula omitted

We give a new proof that there exists a unique strongly regular graph with parameters ( 81 , 20 , 1 , 6 ) . Unlike the finite geometry approach used by Brouwer and Haemers, we use linear algebra and spectral graph theory concepts, namely the technique of star complements, in our proof.

Motivating the concept of eigenvectors via cryptography

New methods of teaching linear algebra in the undergraduate curriculum have attracted much interest lately. Most of this work is focused on evaluating and discussing the integration of special computer software into the Linear Algebra curriculum. In this article, I discuss my approach on introducing the concept of eigenvectors and eigenvalues, which I have used for the last 3 years in my Linear Algebra course. I offer some examples on how I have attracted the interest of our students via Hill ciphering, a method of cryptography. After emphasizing the effect of a linear transformation in a vector space and the importance of eigenvectors, I show how students’ motivation and understanding towards one of the abstract concepts in Linear Algebra; eigenvalues and eigenvectors have grown positively.

Noncentral Chi-Square Versus Normal Distributions in Describing the Likelihood Ratio Statistic: The Univariate Case and Its Multivariate Implication

In the literature of mean and covariance structure analysis, noncentral chi-square distribution is commonly used to describe the behavior of the likelihood ratio (LR) statistic under alternative hypothesis. Due to the inaccessibility of the rather technical literature for the distribution of the LR statistic, it is widely believed that the noncentral chi-square distribution is justified by statistical theory. Actually, when the null hypothesis is not trivially violated, the noncentral chi-square distribution cannot describe the LR statistic well even when data are normally distributed and the sample size is large. Using the one-dimensional case, this article provides the details showing that the LR statistic asymptotically follows a normal distribution, which also leads to an asymptotically correct confidence interval for the discrepancy between the null hypothesis/model and the population. For each one-dimensional result, the corresponding results in the higher dimensional case are pointed out and references are provided. Examples with real data illustrate the difference between the noncentral chi-square distribution and the normal distribution. Monte Carlo results compare the strength of the normal distribution against that of the noncentral chi-square distribution. The implication to data analysis is discussed whenever relevant. The development is built upon the concepts of basic calculous, linear algebra, and introductory probability and statistics. The aim is to provide the least technical material for quantitative graduate students in social science to understand the condition and limitation of the noncentral chi-square distribution.

Adjoints of a matrix

A set of adjoints for a matrix is defined. Basic concepts in linear algebra and matrix analysis, such as the most important inverses of a matrix and the differential of a determinant, are expressed by adjoints.

Jordan Canonical Form: Application to Differential Equations

Jordan Canonical Form (JCF) is one of the most important, and useful, concepts in linear algebra. In this book we develop JCF and show how to apply it to solving systems of differential equations. We

Embodied, symbolic and formal thinking in linear algebra

Students often find their first university linear algebra experience very challenging. While coping with procedural aspects of the subject, solving linear systems and manipulating matrices, they may struggle with crucial conceptual ideas underpinning them, making it very difficult to progress in more advanced courses. This research has sought to apply Tall's three worlds of mathematics, embodied, symbolic and formal, to the linear algebra concepts of linear combination and linear independence by groups of second year university students. The results suggest that the students struggled to understand the concepts through definitions and primarily process conceptions, but that embodied, visual ideas proved a valuable adjunct to their thinking.

Probabilistic Analysis of a Table Tennis Game

A game of table tennis is analyzed using elementary concepts of calculus, probability, and linear algebra. The analysis includes modeling using different approaches and a comparison of the results obtained under the new and the old rules for the game. The analysis demonstrates the steps of a mathematic modeling and illustrates the possible expansion of it.

Discussion on: “Measurable Signal Decoupling with Dynamic Feedforward Compensation and Unknown-input Observation for Systems with Direct Feedthrough”

In the last four decades, ever since the seminal work of Basile and Marro [1] and Wonham and Morse [6], research efforts have been devoted to analysis and synthesis problems for linear state space systems within the well-known context of the geometric approach. The central issue in this approach is to use basic linear algebraic concepts such as linear mapping and linear subspace to investigate the structural properties of linear systems in terms of properties of the mappings A, B, C, D, etc., appearing in the system equations. For example, properties such as controllability, observability, stabilizability, and detectability can be expressed and characterized very effectively in terms of subspaces generated by these system mappings. Also, important notions such as system invertibility, system zeroes, minimum phase, weak observabilty, and strong controllability are analysed most naturally using the geometric approach. In addition, a large number of synthesis problems has been studied in this framework, such as pole placement and stabilization, observer design, disturbance decoupling by static state feedback or dynamic measurement feedback, observer design in the presence of unknown inputs, problems of output decoupling, problems of tracking and regulation, input-output decoupling, and decentralized control problems. Starting with the famous textbook by W.M. Wonham [7], a treatment of most of these synthesis problems can be found in the textbooks [2] and [4]. In the eighties, the ’exact’ versions of many of these control synthesis problems were generalized to their ’approximative’ versions, resulting in ’high gain’ feedback design problems such as almost disturbance decoupling, almost input-output decoupling, etc., see [5]. Around that time research attention within the systems and control community had been shifted to a large extend to H2 and H∞ control. However, also in the context of H2 and H∞ control an important role continued to be played by problems of disturbance decoupling and almost disturbance decoupling, see e.g [4]. Although the intensity of research within the geometric approach has become less, there is still an active community that devotes attention to further developing the area. L. Ntogramatzidis, the author of the paper to be discussed in this note, is clearly an exponent of this community. In this note we intend to explain the contribution of the

General Adaptive Neighborhood Image Processing:

The so-called General Adaptive Neighborhood Image Processing (GANIP) approach is presented in a two parts paper dealing respectively with its theoretical and practical aspects. The Adaptive Neighborhood (AN) paradigm allows the building of new image processing transformations using context-dependent analysis. Such operators are no longer spatially invariant, but vary over the whole image with ANs as adaptive operational windows, taking intrinsically into account the local image features. This AN concept is here largely extended, using well-defined mathematical concepts, to that General Adaptive Neighborhood (GAN) in two main ways. Firstly, an analyzing criterion is added within the definition of the ANs in order to consider the radiometric, morphological or geometrical characteristics of the image, allowing a more significant spatial analysis to be addressed. Secondly, general linear image processing frameworks are introduced in the GAN approach, using concepts of abstract linear algebra, so as to develop operators that are consistent with the physical and/or physiological settings of the image to be processed. In this paper, the GANIP approach is more particularly studied in the context of Mathematical Morphology (MM). The structuring elements, required for MM, are substituted by GAN-based structuring elements, fitting to the local contextual details of the studied image. The resulting transforms perform a relevant spatially-adaptive image processing, in an intrinsic manner, that is to say without a priori knowledge needed about the image structures. Moreover, in several important and practical cases, the adaptive morphological operators are connected, which is an overwhelming advantage compared to the usual ones that fail to this property.

Lack of set theory relevant prerequisite knowledge

Many students struggle with college mathematics topics due to a lack of mastery of prerequisite knowledge. Set theory language is one such prerequisite for linear algebra courses. Many students’ mistakes on linear algebra questions reveal a lack of mastery of set theory knowledge. This paper reports the findings of a qualitative analysis of a group of linear algebra students’ mistakes on a set of linear algebra questions. The paper also details an in-time intervention (a pedagogical approach) to enhance students’ understanding of linear algebra concepts through advancing their set theory knowledge. Mathematics teachers can consider similar approaches to address their students’ mistakes.

Lower bounds on zero–one matrices

In this paper we work with pairs of zero–one matrices whose product is the full upper-triangular zero–one matrix. We establish a lower bound for the sum of the non-zero entries in them, using elementary concepts of linear algebra. With this algebraic method we reach the best possible lower bound, since pairs of such matrices may be found whose amount of non-zero entries is exactly our lower bound.

Linear algebra and geometry

This note uses some elementary concepts in linear algebra to prove some well-known results in two- and three-dimensional geometry.

The Role of Proof in Comprehending and Teaching Elementary Linear Algebra.

We describe how elementary Linear Algebra can be taught successfully while introducing students to the concept and practice of ‘mathematical proof’.This is done badly with a sophisticated Definition–Lemma–Proof–Theorem–Proof–Corollary(DLPTPC) approach; badly – since students in elementary Linear Algebra courses have very little experience with proofs and mathematical rigor. Instead, the subjects and concepts of Linear Algebra can be introduced in an exploratory and fundamentally reasoned way. One seemingly successful way to do this is to explore the concept of solvability of linear systems first via the row echelon form (REF). Solvability questions lead to row and column criteria for a REF that can be used repeatedly to: compute subspaces, settle linear (in)dependence, find inverses, perform basis change, compute determinants, analyze eigensystems etc. If these subjects are explained heuristically from the first principles of linear transformations, linear equations, and the REF, students experience the power of a concept–built approach and reap the benefit of deep math understanding. Moreover, early ‘salient point’ proofs lead to an intuitive understanding of ‘math proof’. Once the basic concept of ‘proof’ is ingrained in students, more abstract proofs, even DLPTPC style expositions, on normal matrices, the SVD etc. become accessible and understandable to sophomore students. With the help of this gentle early approach, the concept and construct of a ‘math proof’ becomes firmly embedded in the students' minds and helps with future math courses and general scientific reasoning.

A general analytic approach to the study of the electrodynamic properties of d.c. SQUIDs

A general analytic approach to the study of the dynamical properties of Josephson junctions in d.c. SQUIDs is developed. The periodic properties of the observable electrodynamic quantities are rigorously derived by means of elementary linear algebra concepts and the case of d.c. SQUIDs containing one or two π-junctions is discussed.

Education

Education

Performing Operations with Matrices on Spreadsheets

Although created mainly for other purposes, spreadsheets appear to be useful in mathematics education. For instance, Russell (1992) writes about spreadsheet activities in middle school mathematics in his book with the same title. In a Mathematics Teacher article, Hunt (1995) describes how he uses spreadsheets to teach students synthetic substitution, synthetic division, and Newton's method. Our experience shows that the same tool can be useful for performing matru operations and, further on, for introducing students to basic concepts of linear algebra. The interested reader can find additional examples of using spreadsheets in high school mathematics in Sjostrand's (1994) book, which deals with Excel spreadsheets.

Computing Eigenvalues and Eigenvectors Without Determinants

(1998). Computing Eigenvalues and Eigenvectors Without Determinants. Mathematics Magazine: Vol. 71, No. 1, pp. 24-33.