Computation of Eigenvalues and Eigenvectors
Some software packages provide for the calculation of eigenvalues and eigenvectors of matrices directly. The software used for many of the computations in this book is the Student Version of Program CC,3 which does not do this. However, these important system properties can readily be obtained from it, as shown in the following.
Program CC is oriented to the calculation of transfer functions and presents them in various forms; one is the pole-zero form. Any transfer function of the system, for example, that from elevator angle to pitch rate, when displayed in this form, will show the eigenvalues in the denominator. That is how we obtained the eigenvalues presented in Chap. 6.
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For the eigenvectors, we turn to the expansion theorem (A.2,10). Consider a case where the input to the system is 8C = 8(f), Dirac’s delta function. The response of the ith component of the state vector to this input in the mode corresponding to eigenvalue A is
The ratio of this component to x, for the same input 8(f) is
x,(t) = A,(A) *,(0 A, (A)
This ratio gives the ith component of the eigenvector for the mode associated with A. Any component can be chosen for reference instead of xl5 as illustrated in Figs. 6.3 and 6.15.