MATLAB eigenvalues

After the assembling of the stochastic matrix Pd we have to solve the associated non-selfadjoint eigenvalue problem. Our present numerical results have been computed using the code speig by Radke AND S0RENSEN in Matlab,... 

We can check with MATLAB that the model matrix A has eigenvalues -0.29, -0.69, and -10.02. They are identical with the closed-loop poles. Given a block diagram, MATLAB can put the state space model together for us easily. To do that, we need to learn some closed-loop MATLAB functions, and we will defer this illustration to MATLAB Session 5. 

With eigenvalues selected at -9, we have chosen the estimator to be faster than the state feedback, and all the errors are to decay exponentially. We ll make use of the Ackermann s formula in Eq. (9-39) for observer gains. The MATLAB statements are ... 

In addition, we should beware that the indexing of state variables in MATLAB is in reverse order of textbook examples. Despite these differences, the inherent properties of the model remain identical. The most important of all is to check the eigenvalues ... 

The eigenvalues of S are calculated by MatLab to form the diagonal matrix A as... 

Using Matlab, the eigenvalues of this symmetric matrix have been found to be, in decreasing order, 4.9738,0.0253,0.0009, and 0. The corresponding eigenvector matrix Fis... 

In the above example, equation (5.34), it is relatively straightforward to determine the eigenvalues of K. In the example (5.35) it is much more difficult to develop the equations. The Symbolic Toolbox of Matlab can be employed for the task. 

Due to the special structure of MATLAB, readers should be familiar with the mathematical concepts pertaining to matrices, such as systems of linear equations, Gaussian elimination, size and rank of a matrix, matrix eigenvalues, basis change in n-dimensional space, matrix transpose, etc. For those who need a refresher on these topics there is a concise Appendix on linear algebra and matrices at the end of the book. 

To illustrate we first verify the identical behavior of the MATLAB QR based polynomial-root finder roots and MATLAB s QR based matrix eigenvalue finder eig for p s companion matrix P = C(p) First we define p by its coefficient vector in MATLAB s workspace, then we invoke the MATLAB polynomial-root finder roots, followed by its matrix eigenvalue finder eig on the companion matrix ofp. Finally we display the companion matrix P of p. As an example we use p(x) = x3 — 2.x2 + 4 here and represent p by its coefficient vector [1 -2 0 4] in the following line of MATLAB commands. 

Hint Linearize the equations and find the associated matrix eigenvalues by using the built-in MATLAB function eig.]... 

MATLAB Example 4.1 function to give sorted eigenvectors and eigenvalues... 

Using the MATLAB function called re anal.m, calculate values for each column shown in Table 4.1. Use the sample data file called pea dat . Use the results to make and interpret plots of the eigenvalues and Malinowski s RE and REV functions. 

Faber, K., On solving generalized eigenvalue problems using MATLAB,./. Chemom., 11, 87-91, 1997. 

Below a Matlab script for the calculation of a quadrature approximation of order N from a known set of moments iti using the Wheeler algorithm is reported. The script computes the intermediate coefficients sigma and the jacobi matrix, and, as for the PD algorithm, determines the nodes and weights of the quadrature approximation from the eigenvalues and eigenvectors of the matrix. 

There are a number of standard methods to solve the matrix eigenvalue problem. We refer to the MATLAB-programs. A number of scientists have developed effective methods, which are useful for practical purposes, such as large molecules. It should also be mentioned that reliable computer programs exist in most cases. 

The stability criterion for state-space models indicates that stability is determined by the eigenvalues of A. They can be calculated using the MATLAB command, eig, after defining A ... 

MATLAB has its own function, roots.m, for calculating all the roots of a polynomial equation of the form in Eq. (1.11). This function accomplishes the task of finding the roots of the polynomial equation [Eq. (1.11)] by first converting the polynomial to the companion matrix A shown in Eq. (1.56), It then uses the built-in function eig.m, which calculates the eigenvalues of a matrix, to evaluate the eigenvalues of the companion matrix, which are also the roots of the polynomial Eq. (1.11) ... 

The eigenvalues and eigenvectors of matrix A can be calculated using the techniques developed in Chap. 2 or simply by applying the built-in function eig in MATLAB. This is demonstrated in Example 5.2. 

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