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Vector Spaces
Published in James K. Peterson, Basic Analysis II, 2020
The roots to the characteristic equation are, of course, the eigenvalues of the coefficient matrix A. We usually organize the eigenvalues with the largest one first, although we don’t have to. In fact, in the examples, we organize from small to large! Example: The eigenvalues are −2 and −1. So r1= −1 and r2= −2. Since e−2t decays faster than e−t, we say the root r1= −1 is the dominant part of the solution.Example: The eigenvalues are −2 and 3. So r1= 3 and r2= −2. Since e−2t decays and e3t grows, we say the root r1= 3 is the dominant part of the solution.Example: The eigenvalues are 2 and 3. So r1= 3 and r2= 2. Since e2t grows slower than e3t, we say the root r1= 3 is the dominant part of the solution. For each eigenvalue r we want to find nonzero vectors V so that (rI −A)V = 0 where to help with our writing we let 0 be the two dimensional zero vector. The nonzero V are called the eigenvectors for eigenvaluer and satisfy AV = rV.
Root Locus Design Methods
Published in Arthur G.O. Mutambara, Design and Analysis of Control Systems, 2017
The characteristic equation of a system is based on the transfer function that models the system. There is only one characteristic equation for a given system. The characteristic equation is defined by equating the denominator of the transfer function to zero. Hence, for the system in Figure 5.1 the characteristic equation is given by a(s) = 0 and is thus obtained from
Linear Algebra
Published in Russell L. Herman, A Course in Mathematical Methods for Physicists, 2013
This is the same equation as the characteristic equation for the general constant coefficient differential equation considered in the previous chapter. Thus, the eigenvalues correspond to the solutions of the characteristic equation for the system.
Spectrum of the free rod under tension and compression
Published in Applicable Analysis, 2019
L. Mercredi Chasman, Jooyeon Chung
In this section, we treat the case of eigenvalue branches lying in the upper half-plane . Recall that the eigenvalue equation has the form ; then the characteristic equation is . As we will see, the upper half-plane corresponds to the case that the characteristic equation has real and purely-imaginary complex roots. We will identify the eigenfunctions, provide a complete description for the eigenvalues as functions of via parameterization, and identify some key properties of the eigenvalue curves.
On piezoelectric effect based on Green–Lindsay theory of thermoelasticity
Published in Waves in Random and Complex Media, 2021
Saroj Mandal, Smita Pal(Sarkar)
For the matrix , the characteristic equation is , i.e. with the eigen values ; Thus the matrix exponential is As 's satisfy Equation (36). Hence we have Solving for and yields as Therefore is given by the following matrix According to boundary conditions at x = 0 it is observed that , i.e. . Thus it follows from Equation (34) that where is obtained as here which are obtained using the other boundary conditions and after simplification the required solutions are obtained as follows where ; .
Surface waves in piezoelectric semiconductor by using Eigen value approach
Published in Waves in Random and Complex Media, 2023
Adnan Jahangir, Sayed M. Abo-Dahab, Aiman Iqbal
The roots of characteristic equation are the required Eigen values . The eigenvector accordingly to the eigen value are , the constant values of these Eigen vectors are represented in Appendix-I, whereas