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## Example

We continue to use the example introduced in §2.1 and Figure 2.1. We now consider the case where there are arbitrary positive masses , but the damping constants are zero. This simplifies the equations of motion to . We again solve them by substituting , where is a constant vector and is a constant scalar to be determined. This yields

Thus is an eigenvector and is an eigenvalue of the generalized Hermitian eigenproblem . Since and are positive definite, the eigenvalues are positive, so is pure imaginary and we find that is periodic with period .

Following item 3 in §2.3.7, we may convert this to a standard Hermitian eigenvalue problem as follows. Let be the Cholesky decomposition of . Thus is simply the diagonal matrix . Then the eigenvalues of are the same as the eigenvalues of the symmetric tridiagonal matrix

 (2)

Symmetric tridiagonal matrices have particularly fast and efficient eigenvalue algorithms.

Next: Singular Value Decomposition  J. Up: Generalized Hermitian Eigenproblems   Previous: Related Eigenproblems   Contents   Index
Susan Blackford 2000-11-20