Finding normal modes of a loaded string with the method of Lagrange multipliers

We consider the normal mode problem of a vibrating string loaded with n identical beads of equal spacing, which involves an eigenvalue problem. Unlike the conventional approach to solving this problem by considering the difference equation for the components of the eigenvector, we modify the eigenvalue equation by introducing matrix-valued Lagrange undetermined multipliers, which regularize the secular equation and make the eigenvalue equation non-singular. Then, the eigenvector can be obtained from the regularized eigenvalue equation by multiplying the indeterminate eigenvalue equation by the inverse matrix. We find that the inverse matrix is nothing but the adjugate matrix of the original matrix in the secular determinant up to the determinant of the regularized matrix in the limit that the constraint equation vanishes. The components of the adjugate matrix can be represented in simple factorized forms. Finally, one can directly read off the eigenvector from the adjugate matrix. We expect this new method to be applicable to other eigenvalue problems involving more general forms of the tridiagonal matrices that appear in classical mechanics or quantum physics.

On solvability of the Dirichlet and Neumann boundary value problems for the Poisson equation with multiple involution

Transformations of the involution type are considered in the space $R^l$, $l\geq 2$. The matrix properties of these transformations are investigated. The structure of the matrix under consideration is determined and it is proved that the matrix of these transformations is determined by the elements of the first row. Also, the symmetry of the matrix under study is proved. In addition, the eigenvectors and eigenvalues of the matrix under consideration are found explicitly. The inverse matrix is also found and it is proved that the inverse matrix has the same structure as the main matrix. The properties of the nonlocal analogue of the Laplace operator are introduced and studied as applications of the transformations under consideration. For the corresponding nonlocal Poisson equation in the unit ball, the solvability of the Dirichlet and Neumann boundary value problems is investigated. A theorem on the unique solvability of the Dirichlet problem is proved, an explicit form of the Green's function and an integral representation of the solution are constructed, and the order of smoothness of the solution of the problem in the Hölder class is found. Necessary and sufficient conditions for the solvability of the Neumann problem, an explicit form of the Green's function, and the integral representation are also found.

Description of the Markov chain using the producing function

In article the way of creation of an assumed function of transformation with the help of a generating function and also a method of use of characteristic numbers and vectors for creation of a matrix which elements well describe conditions of process at any moment is described. This approach differs from preceding that, having used a concept of characteristic numbers and characteristic vectors of a matrix of the transitional probabilities, it is possible to simplify considerably calculation of the elements characterizing process. In article methods of stochastic model operation, ways of the description of a generating function, the solution of matrixes of the equations by means of characteristic numbers and vectors are used. Using properties of a generating function, made “dictionary” of z-transformations which helped to define an assumed function of transformation. The generating function of a vector was applied to a research of behavior of a vector of absolute probabilities which elements represent stationary probabilities. For definition of degree of a matrix of transition of probabilities used a concept of characteristic numbers and characteristic vectors of the transitional probabilities. Determined by such way an unlimited set of latent vectors of which made matrixes which describe a condition of a system at any moment. Reception of definition of latent vectors in more difficult examples which is that along with required coefficients of secular equations the system of auxiliary matrixes and an inverse matrix is under construction is also described.

Inverse properties of a class of seven-diagonal (near) Toeplitz matrices

This paper presents the explicit inverse of a class of seven-diagonal (near) Toeplitz matrices, which arises in the numerical solutions of nonlinear fourth-order differential equation with a finite difference method. A non-recurrence explicit inverse formula is derived using the Sherman-Morrison formula. Related to the fixed-point iteration used to solve the differential equation, we show the positivity of the inverse matrix and construct an upper bound for the norms of the inverse matrix, which can be used to predict the convergence of the method.

Aplikasi Operasi Matriks pada Perancangan Simulasi Metode Hill Cipher Menggunakan Microsoft Excel

<p><em>One of several algorithms in cryptography is the Hill Cpher method. This method is used in randomizing the contents of the message using a password key in the form of a matrix with the order MXM. The product in this research is a simulation design to explain how a plaintext message is converted into cipher text with encryption and decryption processes and vice versa from ciphertext to paintext through multiplication and inverse matrix operations. using Microsoft Excel (MS Excel). In this study, a matrix with order 2x2 and modulo 26 was used. This simulation design shows how the work process visually for each step in the process of changing the message content in the Hill Cipher method. It is hoped that the results of this study can be used as a learning medium by the community, especially students and lecturers, to increase understanding and broaden their horizons about the Hill Cipher method in cryptography lectures.</em><em></em></p>

Analisis Kesalahan Siswa Berdasarkan Objek Matematika Menurut Soedjadi pada Materi Determinan dan Invers Matriks

AbstrakSebagian besar siswa terkadang membuat kesalahan dalam menyelesaikan soal-soal matematika baik yang disengaja maupun tidak disengaja. Penelitian ini bertujuan untuk menganalisis kesalahan-kesalahan yang dilakukan siswa dalam mengerjakan soal pada materi determinan dan invers matriks. Jenis penelitian ini adalah penelitian kualitatif. Teknik pengumpulan data yang digunakan adalah teknik tes dan wawancara secara daring. Subjek penelitian yaitu 30 siswa XI MIA 1 MAN 3 Kota Pekanbaru tahun pelajaran 2020/2021. Analisis kesalahan siswa dilihat berdasarkan objek matematika menurut Soedjadi yaitu fakta, konsep, prinsip, dan operasi. Hasil analisis kesalahan menunjukkan bahwa kesalahan paling banyak dilakukan siswa adalah kesalahan konsep dengan persentase sebesar 17,3%. Penyebab terjadinya kesalahan yang dilakukan siswa adalah siswa belum memahami konsep matriks, siswa lupa dengan konsep matriks dan kurang teliti dalam melakukan operasi perhitungan. Dalam pembelajaran, hendaknya guru tidak mengajarkan siswa untuk menghafalkan rumus namun lebih mengutamakan pemahaman konsep siswa. AbstractMost students sometimes make mistakes in solving math problems, either deliberately or unintentionally. This study aims to analyze the errors made by students in working on the questions on the determinant and inverse matrix material. This type of research is qualitative research. The data collection techniques used were online test and interview techniques. The research subjects were 30 students of XI MIA 1 MAN 3 Pekanbaru City in the 2020/2021 school year. Analysis of student errors is seen based on mathematical objects according to Soedjadi, namely facts, concepts, principles, and operations. The results of the error analysis showed that the most mistakes made by students were misconceptions with a percentage of 17.3%. The cause of the errors made by students is that students do not understand the concept of the matrix, students forget the concept of the matrix, and are not careful in performing calculation operations. In learning, the teacher should not teach students to memorize formulas but prioritize students' understanding of concepts.