Eigenvalue
Eigenvalue is a fundamental concept in linear algebra and is used in various technological and engineering applications. It represents a scalar value that characterizes the behavior of a linear transformation or a matrix. Eigenvalues are crucial for understanding stability, oscillations, and equilibrium points in systems, making them essential for analyzing and solving engineering problems.
5 Key excerpts on "Eigenvalue"
eBook - ePub- Wen Geyi(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...3 Eigenvalue Problems The validity of theorems on eigenfunctions can be made plausible by the following observation made by Daniel Bernoulli (1700–1782). A mechanical system of n degrees of freedom possesses exactly n eigensolutions. A membrane is, however, a system with an infinite number of degrees of freedom. This system will, therefore, have an infinite number of eigenoscillations. —Arnold Sommerfeld In 1894, the French mathematician Jules Henri Poincare´ (1854–1912) established the existence of an infinite sequence of Eigenvalues and the corresponding eigenfunctions for the Laplace operator under Dirichlet boundary condition. This key result signifies the beginning of spectral theory, which extends the eigenvector and Eigenvalue theory of a square matrix and has played an important role in mathematics and physics. The study of Eigenvalue problems has its roots in the method of separation of variables. An eigenmode of a system is a possible state when the system is free of excitation, which might exist in the system on its own under certain conditions, and is also called an eigenstate of the system. The corresponding Eigenvalue often represents an important quantity of the system, for example the total energy of the system and the natural oscillation frequency. The eigenmode analysis has been extensively used in physics and engineering science as an arbitrary state of the system can be expressed as a linear combination of the eigenmodes. When the Eigenvalue problem is solved, what remains is to determine the expansion coefficients in the linear combination by using the source conditions or the initial values of the system. In most situations, only one or a few eigenmodes dominate in the linear combination. The electromagnetic Eigenvalue problems are often expressed by differential equations. The corresponding differential operators are typically positive-bounded-below and symmetric if the medium is isotropic and homogeneous...
eBook - ePub- Michael Harrison, Patrick Waldron(Authors)
- 2011(Publication Date)
- Routledge(Publisher)
...Eigenvalues and eigenvectors DOI: 10.4324/9780203829998-5 3.1 Introduction Some of the basic ideas and issues encountered in the previous chapters are often covered in an introductory course in mathematics for economics and finance. The fundamental ideas of Eigenvalues and eigenvectors and the associated theorems introduced in this chapter are probably not. Many readers are therefore likely to be encountering these concepts for the first time. Hence this chapter begins by providing definitions and illustrations of Eigenvalues and eigenvectors, and explaining how they can be calculated. It goes on to examine some of the uses of these concepts and to establish a number of theorems relating to them that will be useful when we return to the detailed analysis of our various applications. 3.2 Definitions and illustration Eigenvalues and eigenvectors arise in determining solutions to equations of the form A x = λ x (3.1) where A is an n × n matrix, x is a non-zero n -vector and λ is a scalar, and where the solution is for λ and x, given A. We shall call equations like (3.1) eigenequations. The scalar λ is called an Eigenvalue of A, while x is known as an eigenvector of A associated with λ. Sometimes the value, λ, and the vector, x, are called the proper, characteristic or latent value and vector. Consider the matrix A = [ 2 0 8 − 2 ] and the vector x = [ 1 2 ]. Since A x = [ 2 0 8 − 2 ] [ 1 2 ] = [ 2 4 ] = 2 x (3.2) λ = 2 is an Eigenvalue of A and x is an associated eigenvector. It is easy to check, by substituting into the eigenequation (3.1), that another eigenvector of A associated with λ = 2 is [−1 − 2] ┬. Likewise, another Eigenvalue of A is −2, which has associated with it eigenvectors such as [0 1] ┬ and [0 − 1] ┬. Thus, for a given λ, we note that there are multiple associated eigenvectors. For given λ and x, A may be viewed as the matrix that, by pre-multiplication, changes all of the elements of x by the same proportion, λ...
eBook - ePubSteel Structures
Design using FEM
- Rolf Kindmann, Matthias Kraus(Authors)
- 2012(Publication Date)
- Ernst & Sohn(Publisher)
...The terms Eigenvalue, eigenvector and eigenmode are standard mathematical language. Instead of Eigenvalue, the terms critical load and critical stress are used by engineers. Eigenmodes being described through their eigenvectors are called “buckling shapes” for member and plate buckling. If Eq. (6.31) – which is, to be more precise, actually a condition – is formulated in words, the following task can be stated for buckling: the lowest critical load factor η cr and all values of the eigenvector v describing the buckling shape have to be determined so that the value zero occurs in each row after the execution of the calculation operations of Eq. (6.31). As far as the determination of the critical load is concerned, we often don’t bother solving the condition in formula (6.31) but use the following condition instead: (6.32) The determination of Eigenvalues and eigenmodes is an extremely challenging task, both mathematically and from the engineering viewpoint. A look at the relevant literature shows that many solution methods exist. In many cases, it is not directly noticeable which method is appropriate or practical for a given task. Since they are all iterative procedures, the convergence behaviour is of vital importance. After a calculation, we always need to ask whether the required Eigenvalue and associated eigenmode really have been found. The evaluation is fairly difficult and therefore it requires some experience. In the following Section, some explanations will be given and then two methods, the matrix decomposition procedure and the inverse vector iteration, will be dealt with in detail. 6.2.2 Explanations for Understanding Often one must judge whether the calculation results of computer programs are correct or not...
eBook - ePubRobust Control System Design
Advanced State Space Techniques
- Chia-Chi Tsui(Author)
- 2022(Publication Date)
- CRC Press(Publisher)
...It should be emphasized again that based on the analysis of Chapter 2, Eigenvalues indicate far more accurately system performance than any other system parameters such as bandwidth, and the sensitivity of an Eigenvalue is determined by its eigenvectors. Therefore, Eigenvalue and eigenvector assignment can improve system performance and robustness far more directly and effectively, than other design objectives. 8.1 Eigenvalue (Pole) Selection Although system poles most directly determine system performance, there are no uniquely and generally optimal rules for feedback system pole selection. This is because plant system conditions are very diverse and very complicated, and because the performance and robustness design requirements are contradictory to each other. Therefore, it is impossible to have general, explicit, and really optimal pole selection rules for all system conditions and all design requirements, and it is also impossible to have a really good pole selection without trial and error. Nonetheless, there are still some basic and general understandings about the relationship between the system poles and the system performance and robustness (see Figure 2.1 and Conclusion 2.2). The following six general and explicit pole selection rules are guided by these basic understandings (Truxal, 1955). The more negative the real part of the poles, the faster the system transient response before reaches its steady state. In regulator problems, it is often required that the zero-frequency response of the control system T (s = 0) be a specific constant. For example, if the unit-step response y (t) is required to reach a constant 1 as t →∞, then according to the final value theorem that y (t →∞) = sY (s → 0) = sT (s →0)/ s = T (s →0) = 1...
eBook - ePub- James Davidson(Author)
- 2018(Publication Date)
- Wiley(Publisher)
...9 Eigenvalues and Eigenvectors This chapter represents something of a change of gear in the study of matrices. So far, we have learned how to do various calculations using matrix notation and exploit the fairly simple rules of matrix algebra. Most importantly, we have used linearity to calculate the expected values of matrix functions of random variables and so computed means and variances of estimators. However, all these calculations could in principle have been done using ‘sigma’ notation with scalar quantities. Matrix algebra simply confers the benefits of simplicity and economy in what would otherwise be seriously complicated calculations. What happens in this chapter is different in kind because the methods don't really have any counterparts in scalar algebra. One enters a novel and rather magical world where seemingly intractable problems turn out to have feasible solutions. Careful attention to detail will be necessary to keep abreast of the new ideas. This is a relatively technical chapter, and some of the arguments are quite intricate. This material does not all need to be completely absorbed in order to make use of the results in the chapters to come, and readers who are happy to take such results on trust may want to skip or browse it at first reading. The material essential for understanding least squares inference relates to the diagonalization of symmetric idempotent matrices. 9.1 The Characteristic Equation Let be square, and consider for a scalar the scalar equation (9.1) This is the determinant of the matrix formed from by subtracting from each of its diagonal elements. It is known as the characteristic equation of. By considering the form of the determinant, as in (3.8), this is found to be a ‐order polynomial in. The determinant is a sum of terms, of which one is the product of the diagonal elements. When multiplied out, it must contain the term, and in general, powers of of all orders up to appear in one term or another...
