scholarly journals Can One Hear a Matrix? Recovering a Real Symmetric Matrix from Its Spectral Data

Author(s):  
Tomasz Macia̧żek ◽  
Uzy Smilansky

AbstractThe spectrum of a real and symmetric $$N\times N$$ N × N matrix determines the matrix up to unitary equivalence. More spectral data is needed together with some sign indicators to remove the unitary ambiguities. In the first part of this work, we specify the spectral and sign information required for a unique reconstruction of general matrices. More specifically, the spectral information consists of the spectra of the N nested main minors of the original matrix of the sizes $$1,2,\ldots ,N$$ 1 , 2 , … , N . However, due to the complicated nature of the required sign data, improvements are needed in order to make the reconstruction procedure feasible. With this in mind, the second part is restricted to banded matrices where the amount of spectral data exceeds the number of the unknown matrix entries. It is shown that one can take advantage of this redundancy to guarantee unique reconstruction of generic matrices; in other words, this subset of matrices is open, dense and of full measure in the set of real, symmetric and banded matrices. It is shown that one can optimize the ratio between redundancy and genericity by using the freedom of choice of the spectral information input. We demonstrate our constructions in detail for pentadiagonal matrices.

2021 ◽  
Vol 30 (3) ◽  
pp. 59-75
Author(s):  
M. A. Golovchin

In 2016-2018 the state in Russia adopted a package of program documents, which implies the transfer of education to the large-scale introduction of digital technologies. This phenomenon has been called “digitalization of education”. In scientific literature, electronization and digitalization are increasingly called one of the institutional traps for the development of Russian universities, since the corresponding institutional environment has not yet been formed due to the forced nature of innovations. As a result, the processes of introducing new technologies into education are still not regulated. Within the framework of the purpose of the study, the manifestations of the trap of electronization and digitalization of Russian higher education were analyzed on the basis of sociological data, and the theoretical modeling of the process of adaptation of educational agents to the institution of digitalization was carried out.In the course of the study, the approaches were summarized that have been developed in discussions on educational digitalization. The article presents the author’s vision of the studied phenomenon as an institutional trap; as well as understanding of the institutional features and characteristics of electronization and digitalization in education.The research method is the analysis of estimates obtained in the course of an expert survey which was conducted by the Vologda Scientific Center of the Russian Academy of Sciences among the representatives of the teaching staff of state universities in the Vologda region. In the course of this analysis, the indicators of educational digitalization as an effective innovation were clarified such as an increased accessibility of educational resources; simplification of communication and the process of transferring knowledge from teacher to student; increased opportunities for training specialists for the new (digital) economy; improving the quality of education in universities, etc. Based on the results of the empirical study, it has been determined that the conditions for the development of digitalization in Russian universities are currently ambiguous, which is closely related to the level of competitiveness of the educational organization.The scientific novelty of the research consists in the presentation of an original matrix describing the process of university employees adaptation to the conditions of digital transformation of education. The matrix is proposed on the basis of a sociological analysis of the impact of the trap of electronization and digitalization on the activities of educational agents. The matrix can be taken into account in the practice of higher education management.


2007 ◽  
Vol 14 (01) ◽  
pp. 97-102 ◽  
Author(s):  
Q. Mushtaq ◽  
U. Hayat

We show that the matrix A(g), representing the element g = ((xy)2(xy2)2)m (m ≥ 1) of the modular group PSL(2,Z) = 〈x,y : x2 = y3 = 1〉, where [Formula: see text] and [Formula: see text], is a 2 × 2 symmetric matrix whose entries are Pell numbers and whose trace is a Pell–Lucas number. If g fixes elements of [Formula: see text], where d is a square-free positive number, on the circuit of the coset diagram, then d = 2 and there are only four pairs of ambiguous numbers on the circuit.


Author(s):  
Vladimir N. Lutay

The solution of systems of linear algebraic equations, which matrices can be poorly conditioned or singular is considered. As a solution method, the original matrix is decomposed into triangular components by Gauss or Chole-sky with an additional operation, which consists in increasing the small or zero diagonal terms of triangular matrices during the decomposition process. In the first case, the scalar products calculated during decomposition are divided into two positive numbers such that the first is greater than the second, and their sum is equal to the original one. In further operations, the first number replaces the scalar product, as a result of which the value of the diagonal term increases, and the second number is stored and used after the decomposition process is completed to correct the result of calculations. This operation increases the diagonal elements of triangular matrices and prevents the appearance of very small numbers in the Gauss method and a negative root expression in the Cholesky method. If the matrix is singular, then the calculated diagonal element is zero, and an arbitrary positive number is added to it. This allows you to complete the decomposition process and calculate the pseudo-inverse matrix using the Greville method. The results of computational experiments are presented.


Author(s):  
Д.А. МЕТЛЕНКИН ◽  
Ю.Т. ПЛАТОВ ◽  
Р.А. ПЛАТОВА ◽  
А.Е. РУБЦОВ ◽  
А.М. МИХАЙЛОВА

Для идентификации кофе используют методы газовой и жидкостной хроматографии, которые дают точную и подробную информацию о его химическом составе, однако трудоемки, сложны по пробоподготовке и непригодны для оперативного мониторинга качества. Цель настоящего исследования – разработка и апробация метода идентификации кофе по ботаническому виду, географическому месту произрастания и обжарке с применением Фурье-ИК-спектроскопии и многомерного анализа. В качестве объектов исследования были образцы кофе в зернах, различающиеся по ботаническому виду (арабика/робуста), географическому месту произрастания (Азия/Америка/Африка) и обжарке (жареный/нежареный). Для разработки моделей идентификации кофе в зернах была сформирована база спектральных данных и применены методы многомерного анализа – метод главных компонент (МГК) и дискриминантный анализ (ДА). ИК-спектры образцов кофе регистрировали с помощью Фурье-ИК-спектрометра Bruker ALPHA с алмазным модулем НПВО в диапазоне 4000–400 см–1 при разрешающей способности спектрометра 2 см–1. Спектральные данные были экспортированы из встроенного программного обеспечения OPUS 7.3.5.0 в Excel. При анализе матрицы спектральных данных выявлены наиболее интенсивные полосы поглощения ИК-спектра, приписываемые наличию функциональных групп воды, липидов, полисахаридов, кофеина и хлорогеновой кислоты в кофе. При сравнении ИК-спектров образцов кофеина, декофеинизированного кофе и кофе в зернах выявлены полосы поглощения спектра, которые можно использовать для построения калибровочной модели содержания кофеина в составе кофе в зернах. По спектральным данным МГК построена многомерная модель градации образцов кофе в зависимости от ботанического вида и наличия обжарки. По матрице факторных нагрузок выявлены полосы поглощения спектра, объясняющие различия образцов по ботаническому виду и обжарке и вносящие наибольший вклад в разделение образцов кофе на группы. Методом ДА по 19 переменным – коэффициентам поглощения на волновых числах спектра разработана система классификационных функций градации образцов кофе по географическому месту произрастания. Доказано, что сочетание Фурье-ИК-спектроскопии с методами многомерного анализа можно использовать как быстрый и неразрушающий инструмент для идентификации кофе в зернах. Gas and liquid chromatography methods are used to identify coffee. They provide accurate and detailed information about its chemical composition; however they are time-consuming, complex in sample preparation and unsuitable for operational quality monitoring. The purpose of this study is to develop and test a method for identifying coffee by botanical species, geographical place of growth and roasting using FTIR-spectroscopy and multivariate analysis. Samples of coffee beans were selected as objects of research, differing in botanical type (Arabica/Robusta), geographical place of growth (Asia/America/Africa) and roasting (roasted/not roasted). To develop models for the identification of grain coffee, a spectral database was formed and the methods of multivariate analysis were applied: principal components analysis (PCA), discriminant analysis. The IR-spectra of coffee samples were recorded using a Bruker ALPHA FTIR-spectrometer with a diamond module in the range of 4000–400 cm–1 with a resolution of the spectrometer of 2 cm–1. Spectral data were exported from the OPUS 7.3.5.0 embedded software to Excel. During analysis the matrix of spectral data, the most intense absorption bands of the IR-spectrum were revealed, attributed to the presence of functional groups of water, lipids, polysaccharides, caffeine and chlorogenic acid in grain coffee. By comparison the IR spectra of the samples: caffeine, decaffeinated coffee and grain coffee, absorption bands of the spectrum were revealed, which can be used to build a calibration model of the caffeine content in the composition of coffee beans. Using PCA based on the spectral data, a multivariate model of the gradation of coffee by botanical type and depending on the roast was build. According to the matrix of factor loadings, absorption bands of the spectrum were revealed, explaining the differences between the samples in botanical type and roasting and making the greatest contribution to the division of coffee samples into groups. By the method of discriminant analysis using 19 variables – absorption coefficients at the wave numbers of the spectrum – a system of classification functions for the gradation of grain coffee samples according to the geographical place of growth has been developed. It is proved that the combination of FTIR-spectroscopy with multivariate analysis methods can be used as a fast and non-destructive tool for identifying coffee beans.


1978 ◽  
Vol 235 (1) ◽  
pp. R93-R98 ◽  
Author(s):  
D. H. Perkel ◽  
B. Mulloney

Numerical parameters for a compartmental model of a neuron can be chosen to conform both to the neuron's structure and to its measured steady-state electrical properties. A systematic procedure for assigning parameters is described that makes use of the matrix of coefficients of the set of differential equations that embodies the compartmental model. The inverse of this matrix furnishes input resistances and voltage attenuation factors for the model, and an interactive modification of the original matrix and its inverse may be used to fit the model to anatomic and electrical measurements.


1952 ◽  
Vol 48 (1) ◽  
pp. 110-117
Author(s):  
K. J. Le Couteur

AbstractIt is proved that the matrix algebra for any relativistic wave equation of half-odd integral spin can be factorized as the direct product of a Dirac algebra and another, called a ξ-algebra. The structure and representation of ξ-algebras are studied in detail. The factorization simplifies calculations with particles of spin > ½, because the ξ-algebra contains only one-sixteenth as many elements as the original matrix algebra.


1983 ◽  
Vol 66 ◽  
pp. 331-341
Author(s):  
M. Knölker ◽  
M. Stix

AbstractThe differential equations describing stellar oscillations are transformed into an algebraic eigenvalue problem. Frequencies of adiabatic oscillations are obtained as the eigenvalues of a banded real symmetric matrix. We employ the Cowling-approximation, i.e. neglect the Eulerian perturbation of the gravitational potential, and, in order to preserve selfadjointness, require that the Eulerian pressure perturbation vanishes at the outer boundary. For a solar model, comparison of first results with results obtained from a Henyey method shows that the matrix method is convenient, accurate, and fast.


2014 ◽  
Vol 2014 ◽  
pp. 1-10
Author(s):  
Jun-Lin Lin ◽  
Hung-Chjh Chuan ◽  
Laksamee Khomnotai

A system of fuzzy relational equations with the max-Archimedeant-norm composition was considered. The relevant literature indicated that this problem can be reduced to the problem of finding all the irredundant coverings of a binary matrix. A divide-and-conquer approach is proposed to solve this problem and, subsequently, to solve the original problem. This approach was used to analyze the binary matrix and then decompose the matrix into several submatrices such that the irredundant coverings of the original matrix could be constructed using the irredundant coverings of each of these submatrices. This step was performed recursively for each of these submatrices to obtain the irredundant coverings. Finally, once all the irredundant coverings of the original matrix were found, they were easily converted into the minimal solutions of the fuzzy relational equations. Experiments on binary matrices, with the number of irredundant coverings ranging from 24 to 9680, were also performed. The results indicated that, for test matrices that could initially be partitioned into more than one submatrix, this approach reduced the execution time by more than three orders of magnitude. For the other test matrices, this approach was still useful because certain submatrices could be partitioned into more than one submatrix.


1974 ◽  
Vol 10 (2) ◽  
pp. 245-253 ◽  
Author(s):  
A.N. Stokes

In the domain of real symmetric matrices ordered by the positive definiteness criterion, the symmetric matrix Riccati differential equation has the unusual property of preserving the ordering of its solutions as the independent variable changes, Here is is shown that, subject to a continuity restriction, the Riccati equation is unique among comparable equations in possessing this property.


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