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but the eigenvectors are the same.
== Methods of solution ==
For linear elastic problems that are properly set up (no rigid body rotation or translation),
the stiffness and mass matrices and the system in general are [[Positive-definite matrix|positive definite]].
These are the easiest matrices to deal with because the numerical methods commonly
applied are guaranteed to converge to a solution. When all the qualities of the system are
considered:
# Only the smallest eigenvalues and eigenvectors of the lowest modes are desired
# The mass and stiffness matrices are sparse and highly banded
# The system is positive definite
a typical prescription of solution is first to [[tridiagonal]]ize the system using the
[[Lanczos algorithm]]. Next, use the [[QR algorithm]] to find the eigenvectors and eigenvalues of
this tridiagonal system. If inverse iteration is used, the new eigenvalues will
relate to the old by <math> \mu = \frac{1}{\lambda} </math>, while the eigenvectors of the original can
be calculated from those of the tridiagonalized matrix by:
:<math>
[r^{n}] =
[Q] [v^{n}]
</math>
where <math> [r^{n}] </math> is a Ritz vector approximately equal to
the eigenvector of the original system, <math> [Q] </math> is the matrix
of Lanczos vectors, and <math> [v^{n}] </math> is the <math> n^{th} </math> eigenvector
of the tridiagonal matrix.
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