COMPUTATION OF EIGENVALUES AND EIGENVECTORS

For low-order systems, the characteristic determinant can be directly expanded and the characteristic equation (3.3,7) written out. If n < 4, analytical solutions exist for the roots. For large-order systems the eigen­values and eigenvectors are computed from the system matrix A by digital machine methods (refs. 3.3, 3.4). A discussion of these methods and of their recommended spheres of application is beyond the scope of this volume. Suffice it to say that practical methods and computing routines are available in most computation centers for extracting the eigenvalues and eigenvectors for systems of very large order, even for n > 100.

It is worthwhile describing one fairly direct approach to computation of eigenvectors. Consider (3.3,156) as a homogeneous set of scalar equations with lr known and the n components of ur as the unknowns. Now divide through all the equations by any one of the unknowns, say umr, so that there results n equations for (n — 1) ratios и{г/итг. By dropping any one of the equations and transposing the coefficients of umT to the r. h.s., a complete set of (n — 1) equations is obtained for the (n — 1) ratios. These can be solved by any conventional method to yield the ratios of all the components of ufto umr. The equations will of course have complex coefficients for complex eigen­values, and real coefficients for real eigenvalues. This process for a third – order system would go as follows:

bn(kr)ulr + b12(?ir)u2r + b13(kr)u3r = 0

b21(Xr)ulr + b33(Xr)w3r +

^2з(^г)м3г — 0 hl(K)Ulr + bai(K)U2r + b3z(K)U3r = 0

After dividing by u3r and dropping the third equation we get

h – Її °n —

%3r

b21^ + b2

The solution of this set of equations gives the two required ratios in terms of which the eigenvector is [м1г/м3г, 1]. There are two difficulties

associated with this method. The first is that if u3r turns out to be very small relative to ulT and u2r the equations will be ill-conditioned, and a different choice for the component to divide by has to be made. The second is that when X is complex, there are really two sets of equations to be solved for the real and imaginary parts of the ratios.

Clearly the eigenvector corresponding to the conjugate eigenvalue X* will be itself the conjugate of ur, so only one of the pair need be calculated.

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