Elementary-Linear-Algebra

It is well known that unit root limit distributions are sensitive to initial conditions in the distant past. If the distant past initialization is extended to the infinite past, the initial condition dominates the limit theory, producing a faster rate of convergence, a limiting Cauchy distribution for the least squares coefficient, and a limit normal distribution for the t-ratio. This amounts to the tail of the unit root process wagging the dog of the unit root limit theory. These simple results apply in the case of a univariate autoregression with no intercept. The limit theory for vector unit root regression and cointegrating regression is affected but is no longer dominated by infinite past initializations. The latter contribute to the limiting distribution of the least squares estimator and produce a singularity in the limit theory, but do not change the principal rate of convergence. Usual cointegrating regression theory and inference continue to hold in spite of the degeneracy...

We first define the real matrix-variate gamma function, the gamma integral and the gamma density, wherefrom their counterparts in the complex domain are developed. An important particular case of the real matrix-variate gamma density known as the Wishart density is widely utilized in multivariate statistical analysis. Additionally, real and complex matrix-variate type-1 and type-2 beta density functions are defined. Various results pertaining to each of these distributions are then provided. More general structures are considered as well.

We present a global, synthetic and updated vision of the contributions made by the anthropological theory of the didactic to the problem of teaching elementary algebra. We start by summarising the first results obtained by Yves Chevallard in the 80s that have marked the trend of research on elementary algebra in the anthropological approach. A reference epistemological model of elementary algebra is then proposed. This model provides a new interpretation of the role of elementary algebra in current school mathematics and guides the study of its ecology, that is, the constraints which partially explain the current situation and the conditions that could modify it in a given direction.

The new technological approaches bring us into the digital era, where data security is a part of our everyday lives. Nowadays cryptographic algorithms, which are also recommended by international security standards, are often developed for non-limited devices and they are not suitable for limited environments. This paper deals with a lightweight solution for the structure of big numbers, which should help with ordinary recommended cryptographic algorithms in limited devices. We introduce our lightweight structure and elementary algebra for cryptographic primitives based on non-limited OpenSSL library. Last but not least, we provide experimental measurements and verification on the real scenarios.

A new necessary and sufficient condition for the existence of an $m$-th root of a nilpotent matrix in terms of the multiplicities of Jordan blocks is obtained and expressed as a system of linear equations with nonnegative integer entries which is suitable for computer programming. Thus, computation of the Jordan form of the $m$-th power of a nilpotent matrix is reduced to a single matrix multiplication; conversely, the existence of an $m$-th root of a nilpotent matrix is reduced to the existence of a nonnegative integer solution to the corresponding system of linear equations. Further, an erroneous result in the literature on the total number of Jordan blocks of a nilpotent matrix having an $m$-th root is corrected and generalized. Moreover, for a singular matrix having an $m$-th root with a pair of nilpotent Jordan blocks of sizes $s$ and $l$, a new $m$-th root is constructed by replacing that pair by another one of sizes $s+i$ and $l-i$, for special $s,l,i$. This method applies to...

ABSTRACT
The main objective of this action research work was to help improve the understanding of students in SHS 2 in Takoradi Technical Institute in the Western Region of Ghana understanding the concept of solving simultaneous equation in two unknown variables using elimination method.
The population consisted of 396 student of form two, and a sample of 55 students of electricals 2C class. The respondents were 53 boys and 2 girls with an average age of 15years.
The main instruments used to carry out the research were pre-test, post-test and intervention. The pre-test was to find the strength and weakness of the students.
The researcher for the interventions developed strategies by using the guided discovery approach to help students of Takoradi Technical Institute form two overcome their difficulties in solving simultaneous equations in two unknown variables using the elimination method.
After the intervention, the researcher conducted a post- test to evaluate whether the intervention had a positive effect on the student’s understanding of the concept of solving simultaneous equations in two unknown variables by elimination method. The means of the scores were compared and analyzed and the result showed that the student’s performance had improved after the intervention.

A proof of the Jordan canonical form, suitable for a first course in linear algebra, is given. The proof includes the uniqueness of the number and sizes of the Jordan blocks. The value of the customary procedure for finding the block generators is also questioned. ... The Jordan form of linear transformations is an exceeding useful result in all theo- ... Jacobson-Morozov theorem. A classical reference for this topic is Smirnov's book [7, ... Strang's book [8, p.422-425]. The American Mathematical Monthly has published at

The purpose of this to produce efficient numerical methods with the same order of accuracy as that of the main starting values for exact solutions of fourth order differential equation without reducing it to a system of first order differential equations. The methods of the differential systems arising from the approximate solution to the problem are adopted using the Runge-Kutta method and stages. The methods were compared and contrasted based on the results obtained. The comparison shows that Euler method gives accurate approximate result than Runge-Kutta method. After the derivation of the formulae of O(h 2), the comparison was done in regards to identify the formula with higher accuracy.

Peano also postulated the existence of a zero object 0 and used the notation a-b for a + (-b). By introducing the notions of dependent and independent objects, he defined the notion of dimension, showed that finite-dimensional spaces have a basis and gave examples of infinite-dimensional linear spaces. If one considers only functions of degree n, then these functions form a linear system with n + 1 dimensions, the entire functions of arbitrary degree form a linear system with infinitely many dimensions. Peano also introduced linear operators on a linear space and showed that by using coordinates one obtains a matrix. With the passage of time, much concrete has set on these foundations. Techniques and notation have become more refined and the range of applications greatly enlarged. Nowadays Linear Algebra, comprising matrices and vector spaces, plays a major role in the mathematical curriculum. Notwithstanding the fact that many important and powerful computer packages exist to solve problems in linear algebra, it is our contention that a sound knowledge of the basic concepts and techniques is essential.

The matrix, A��� BD��� 1C, is called the Schur Complement of D in M. If A is invertible, then by eliminating x first using the first equation we find that the Schur complement of A in M is D��� CA��� 1B (this corresponds to the Schur complement defined in Boyd and Vandenberghe [1] when C= B).

where β is the vector of regression coefficients, K is the number of levels, X and Σ are square matrices with K − 1 rows/columns, row k of X is the encoding for level k 6= K, and level K is encoded as − ∑K−1 k=1 Xk. This gives an induced prior on the K effects αk = Xkβ that is symmetric in the effects, has a mean of 0 and variance of σ2 for each αk, and is degenerate, in that the sum of the effects is 0. It may be desirable to avoid actually constructing the matrix Σ, either for uniformity in the implementation of a Bayesian regression model (e.g., independent normal priors for each regression coefficient over the entire set of predictor variables) or because K is large. In this note I discuss two ways of achieving this goal.

Symmetric polynomials of the roots of a polynomial can be written as polynomials of the coefficients, and by applying this to the characteristic polynomial we can write a symmetric polynomial of the eigenvalues $a_{i}$ of an $n\times n$ matrix $A$ as a polynomial of the entries of the matrix. We give a magic formula for this: symbolically substitute $a\mapsto A$ in the symmetric polynomial and replace multiplication by $\det$. For instance, for a $2\times2$ matrix $A$ with eigenvalues $a_{1},a_{2}$, \begin{align*} a_1 a_2^2 +a_1^2 a_2 & =\det(A_1, A_2^2)+ \det(A_1^2, A_2) \end{align*} where $A_i^k$ is the $i$-th column of $A^k$. One may also take negative powers, allowing us to calculate: \begin{align*} a_1a_2^{-1}+a_1^{-1}a_{2} & =\det(A_{1},A_{2}^{-1})+\det(A_1^{-1},A_{2}) \end{align*} The magic method also works for multivariate symmetric polynomials of the eigenvalues of a set of commuting matrices, e.g. for $2\times2$ matrices $A$ and $B$ with eigenvalues $a_1,a_2$ and $b_{1},b...

We study certain pairs of subspaces V and W of C^n we call supports that consist of eigenspaces of the eigenvalues ±M of a minimal hermitian matrix M (M≤M+D for all real diagonals D). For any pair of orthogonal subspaces we define a non negative invariant δ called the adequacy to measure how close they are to form a support and to detect one. This function δ is the minimum of another map F defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of F in order to approximate δ. These results allow us to prove that the set of supports has interior points in the space of flag manifolds.

In recent times, the derivation of Runge-Kutta methods based on averages other than the arithmetic mean is on the rise. In this paper, the authors propose a new version of explicit Runge-Kutta method, by introducing the harmonic mean as against the usual arithmetic averages in standard Runge-Kutta schemes.

Abstract. The best method for computing the adjoint matrix of an order n matrix in an arbitrary commutative ring requires O(n log n log log n) operations, provided the complexity of the algorithm for multiplying two matrices is γn + o(n). For a commutative domain – and under the same assumptions – the complexity of the best method is 6γn/(2 − 2) + o(n). In the present work a new method is presented for the computation of the adjoint matrix in a commutative domain. Despite the fact that the number of operations required is now 1.5 times more, than that of the best method, this new method permits a better parallelization of the computational process and may be successfully employed for computations in parallel computational systems.

We study certain pairs of subspaces V and W of C n we call supports that consist of eigenspaces of the eigenvalues ± M of a minimal hermitian matrix M (M ≤ M + D for all real diagonals D). For any pair of orthogonal subspaces we define a non negative invariant δ called the adequacy to measure how close they are to form a support and to detect one. This function δ is the minimum of another map F defined in a product of spheres of hermitian matrices. We study the gradient, Hessian and critical points of F in order to approximate δ. These results allow us to prove that the set of supports has interior points in the space of flag manifolds.

Eigenvalues So far, our applications have concentrated on statics: unchanging equilibrium configurations of physical systems, including mass/spring chains, circuits, and structures, that are modeled by linear systems of algebraic equations. It is now time to set our universe in motion. In general, a dynamical system refers to the (differential) equations governing the temporal behavior of some physical system: mechanical, electrical, chemical, fluid, etc. Our immediate goal is to understand the behavior of the simplest class of linear dynam-ical systems — first order autonomous linear systems of ordinary differential equations. As always, complete analysis of the linear situation is an essential prerequisite to making progress in the vastly more complicated nonlinear realm. We begin with a very quick review of the scalar case, where solutions are simple exponential functions. We use the exponential form of the scalar solution as a template for a possible solution in the vector case. Substituting the resulting formula into the system leads us immediately to the equations defining the so-called eigenvalues and eigenvectors of the coefficient matrix. Further developments will require that we become familiar with the basic theory of eigenvalues and eigenvectors, which prove to be of absolutely fundamental importance in both the mathematical theory and its applications, including numerical algorithms. The present chapter develops the most important properties of eigenvalues and eigen-vectors. The applications to dynamical systems will appear in Chapter 9, while applications to iterative systems and numerical methods is the topic of Chapter 10. Extensions of the eigenvalue concept to differential operators acting on infinite-dimensional function space, of essential importance for solving linear partial differential equations modelling continuous dynamical systems, will be covered in later chapters.Each square matrix has a collection of one or more complex scalars called eigenvalues and associated vectors, called eigenvectors. Viewing the matrix as a linear transformation, the eigenvectors indicate directions of pure stretch and the eigenvalues the degree of stretching. Most matrices are complete, meaning that their (complex) eigenvectors form a basis of the underlying vector space. When expressed in terms of its eigenvector basis, the matrix assumes a very simple diagonal form, and the analysis of its properties becomes extremely simple. A particularly important class are the symmetric matrices, whose eigenvectors form an orthogonal basis of R n ; in fact, this is by far the most common way for orthogonal bases to appear. Incomplete matrices are trickier, and we relegate them and their associated non-diagonal Jordan canonical form to the final section. The numerical computation of eigenvalues and eigenvectors is a challenging issue, and must be be deferred until Section 10.6. Note: Unless you are prepared to consult that section now, solving the computer-based problems in this chapter will require