inverse eigenvalue problem
در نشریات گروه ریاضی-
This paper uses unit lower triangular matrices to solve the nonnegative inverse eigenvalue problem for various sets of real numbers. This problem has remained unsolved for many years for $n \geq 5.$ The inverse of the unit lower triangular matrices can be easily calculated and the matrix similarities are also helpful to be able to solve this important problem to a considerable extent. It is assumed that in the given set of eigenvalues, the number of positive eigenvalues is less than or equal to the number of nonpositive eigenvalues to find a nonnegative matrix such that the given set is its spectrum.Keywords: Nonnegative matrices, unit lower triangular matrices, Inverse eigenvalue problem
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In this paper, for a given set of real interval numbers $\sigma$ that satisfies in special conditions, we find an interval nonnegative matrix $C^I$ such that for each point set $\delta$ of given interval spectrum $\sigma$, there exists a point matrix $C$ of $C^I$ such that $\delta$ is its spectrum. For this purpose, we use unit lower triangular matrices and especially try to use binary unit lower triangular matrices. We also study some conditions for existence solution to the problem.Keywords: Interval matrix, interval arithmetics, inverse eigenvalue problem, nonnegative matrices
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The aim of the current paper is to study a partially described inverse eigenvalue problem of a specific symmetric matrix, and prove some properties of such matrix. The problem includes the construction of the matrix by the minimal eigenvalue of all leading principal submatrices and eigenpair $(\lambda_2^{(n)},x)$ such that $ \lambda_2^{(n)}$ is the maximal eigenvalue of the required matrix. We investigate conditions for the solvability of the problem, and finally an algorithm and its numerical results are presented.Keywords: eigenvalue, eigenpair, leading principal submatrices, Inverse eigenvalue problem
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در این مقاله، برای یک مجموعه داده شده از اعداد حقیقی مانند $ \sigma $ با یک عدد مثبت و مجموع صفر، ما یک ماتریس فاصله می یابیم به گونه ای که مجموعه داده شده طیف ماتریس یافت شده باشد.در نهایت ما حالت خاصی که را حل می کنیم که ماتریس جواب، ماتریس کروی منظم باشد.
کلید واژگان: ماتریس های نامنفی، ماتریس مثلثی واحد، مساله مقدار ویژه معکوس، ماتریس های فاصلهIn this paper, for a given set of real numbers such as $\sigma$ with only one positive number and zero summation, we find a distance matrix in which the given set $\sigma$ is its spectrum.Finally, we solve special cases of the inverse eigenvalue problem in which the matrix solution is a regular spherical distance matrix.
Keywords: Nonnegative matrices, Unit triangular matrices, Inverse eigenvalue problem, Distance matrices -
مقادیر ریتز یک ماتریس، مجموعه کلیه مقادیر ویژه زیرماتریس های اصلی پیش روی آن است. در این مقاله با فرض معلوم بودن مجموعه مقادیر ریتز از بعد حداکثر سه، ما یک ماتریس می یابیم به گونه ای که مجموعه معلوم، مجموعه مقادیر ریتز ماتریس یافت شده باشد. هم چنین شرایط وجود جواب مورد بررسی قرار می گیرد.کلید واژگان: مقادیر ریتز، ماتریس &rlm، سه قطری متقارن، مساله مقدار ویژه معکوسThe Ritz values of a matrix are the set of all the eigenvalues of the leading principal submatrices. In this paper, assuming that the set of Ritz values is given from the dimension of maximum three, we find a matrix such that the given set is its Ritz values. The conditions for the existing solution are also studied.Keywords: Ritz values, Symmetric tridiagonal matrices, Inverse eigenvalue problem
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In this paper, we introduce a new algorithm for constructing a symmetric pentadiagonal matrix by using three interlacing spectrum, say $(\lambda_i)_{i=1}^n$, $(\mu_i)_{i=1}^n$ and $(\nu_i)_{i=1}^n$ such that\begin{eqnarray*}0<\lambda_1<\mu_1<\lambda_2<\mu_2<...<\lambda_n<\mu_n,\\\mu_1<\nu_1<\mu_2<\nu_2<...<\mu_n<\nu_n,\end{eqnarray*}where $(\lambda_i)_{i=1}^n$ are the eigenvalues of pentadiagonal matrix $A$, $(\mu_i)_{i=1}^n$ are the eigenvalues of $A^*$ (the matrix $A^*$ differs from $A$ only in the $(1,1)$ entry) and $(\nu_i)_{i=1}^n$ are the eigenvalues of $A^{**}$ (the matrix $A^{**}$ differs from $A^*$ only in the $(2,2)$ entry). From theinterlacing spectrum, we find the first and second columns of eigenvectors. Sufficient conditions for the solvability of the problem are given. Then we construct the pentadiagonal matrix $A$ from these eigenvectors and given eigenvalues by using the block Lanczos algorithm. We also give an example to demonstrate the efficiency of the algorithm.Keywords: Inverse eigenvalue problem, Pentadiagonal matrix, Interlacing property, Lanczos algorithm
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The problem of constructing a matrix by its spectral information is called inverse eigenvalue problem (IEP) which arises in a variety of applications. In this paper, we study an IEP for arrowhead matrices in different cases. The problem involves constructing of the matrix by some eigenvalues of each of the leading principal submatrices and one eigenpair. We will also investigate this problem and its variants in the cases of matrix entries being real, nonnegative, positive definite, complex and equal diagonal entries. To solve the problems, a new method to establish a relationship between the IEP and properties of symmetric and general form of matrices is developed. The necessary and sufficient conditions of the solvability of the problems are obtained. Finally, some numerical examples are presented.
Keywords: Inverse eigenvalue problem, Arrowhead matrix, Principal submatrix -
In this paper for two given sets of eigenvalues, which one of them is the eigenvalues of circulant matrix and the other is the eigenvalues of skew-circulant matrix, we find a nonnegative matrix, such that the union of two sets be the spectrum of nonnegative matrices.
Keywords: Nonnegative matrices, Circulant, Skew-Circulantmatrices, Inverse eigenvalue problem -
In this paper, the inverse eigenvalue problem for the bordered diagonal matrices are reconsidered whose elements are equal to zero except for the first row, the first column and the diagonal elements. The necessary and sufficient conditions for existence of a symmetric bordered diagonal matrix from special spectral data have been determined. A new algorithm to make such matrices is derived and some numerical examples are given to illustrate the efficiency of the method.Keywords: Inverse eigenvalue problem, bordered diagonal matrix, leading principal submatrix, minimal eigenvalue, maximal eigenvalue
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ارتعاشات سیستم های مختلف مانند جرم-فنر، نخ کشسان، میله و غیره به صورت مسئله مقدار ویژه ماتریس ژاکوبی مدل بندی می شوند. مسئله تعیین ماتریس ژاکوبی با استفاده از داده های طیفی معلوم، مسئله مقدار ویژه معکوس ماتریس ژاکوبی گفته می شود. در این مقاله، ماتریس ژاکوبی با استفاده از دو طیف و یک داده اضافی بازسازی می شود. یکی از طیف ها مقادیر ویژه ماتریس و طیف دیگر مقادیر ویژه زیر ماتریس حاصل از حذف هم زمان دو سطر و ستون ماتریس است. شرایط لازم و کافی روی داده های طیفی را برای حل پذیری مسئله معکوس ارایه کرده و الگوریتم هایی برای تعیین ماتریس ژاکوبی ارایه می گردد. سرانجام با ارایه چند مثال عددی ماتریس ژاکوبی و سیستم جرم- فنر متناظر بازسازی می شوند.
کلید واژگان: مسئله مقدار ویژه معکوس، ماتریس ژاکوبی، داده های طیفی، سیستم جرم-فنرIntroductionMany problems in sciences and engineering can be studied by mathematical models. These models are classified as direct problems and inverse problems. In the structural vibrations, analysis and estimation of the behavior of system e.g. response of the system for an external force and natural frequencies from the known physical parameters is called a direct problem. Determination or estimation of the system physical parameters such as density, mass, stiffness and cross sectional area from the behavior of the system is called an inverse problem. A class of inverse problems which physical parameters determined from the spectral data (eigenvalues, eigenvectors, or both) is called inverse eigenvalue problem. There are many systems such as mass- spring system, vibrating Rods and Beams which are modeled as an eigenvalue problem. Free vibrations of a mass- spring system and discretization of a rod and Sturm-Liouville equations lead to Jacobi matrix eigenvalue problem. Inverse eigenvalue problem for Jacobi matrix is determination of entries using some spectral data. Different algorithms have been presented for constructing a Jacobi matrix. In this paper, we construct a Jacobi matrix and the corresponding mass-spring system using some new spectral data.
Material and methodsWe try to construct a Jacobi matrix from two spectra and one extra data. For this purpose, using given spectral data, we find the required data of well-known Lancsoz method. Then applying Lancsoz method, we construct a positive definite Jacobi matrix. Finally, according to the relations between Jacobi matrix, mass and stiffness matrices, we obtain corresponding mass-spring system.
Results and discussionNecessary and sufficient conditions on given spectral data for solvability of the inverse eigenvalue problem are presented.We find two algorithms for constructing positive definite Jacobi matrix and the corresponding mass-spring system.We solve some examples using the given algorithms. There is a good agreement between the spectral data of constructed matrix and initial given data.
ConclusionThe following results are obtained from this research.We find two algorithms for constructing a Jacobi matrix using two spectra and one extra data.It is observed that, for a set of spectral data, there might be exist more than one solution.It seems that, one may extend the method of this paper for matrix eigenvalue problem which arise in discretization of vibrating rod using finite element method.
Keywords: Inverse eigenvalue problem, Jacobi matrix, Spectral data, Mass-Spring system -
In this paper for a given set of real or complex interval numbers σ satisfying special conditions, we find an interval nonnegative matrix C such that for each point set δ of given interval spectrum σ, there exists a point matrix A of C such that δ is its spectrum. We also study some conditions for the solution existence of the problem.Keywords: Interval matrix, inverse eigenvalue problem, nonnegative matrices
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In this paper, at first for a given set of real or complex numbers $sigma$ with nonnegative summation, we introduce some special conditions that with them there is no nonnegative tridiagonal matrix in which $sigma$ is its spectrum. In continue we present some conditions for existence such nonnegative tridiagonal matrices.Keywords: Inverse eigenvalue problem, Tridiagonal matrix, Nonnegative matrix
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