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Demmel Previous: Invariant Subspaces Contents Index Equivalences (Similarities) Suppose is a nonsingular matrix. Let . We say that is similar to and that is a similarity transformation. has the same eigenvalues as . If is an eigenvector of , so that then is an eigenvector of . , instead of unitary. , we say is unitarily similar to . If , is real, we say orthogonal Susan Blackford 2000-11-20 Next: Conditioning Up: Non-Hermitian Eigenproblems J. Demmel Previous: Equivalences (Similarities) Contents Index Eigendecompositions We discuss four eigendecompositions, or four matrices that are similar to and for which it is simpler to solve eigenproblems.

Demmel Up: Singular Value Decomposition J. 1. We again consider the case with arbitrary masses , and zero damping constants . This simplifies the equations of motion to . 8 we solve the equations of motion by substituting a constant vector and is a constant scalar to be determined. This leads to , where the Cholesky factor is . Letting , we see that we need to compute the eigenvalues of the symmetric tridiagonal matrix Now we note that the stiffness matrix and , where can be factored as Singular Value Decomposition J.

Even though we have effective algorithms for these problems, we cannot necessarily solve all large scale problems in an amount of time and space acceptable to all users. Next: Related Eigenproblems Up: Generalized Hermitian Eigenproblems Previous: Conditioning Contents Index Susan Blackford 2000-11-20 Next: Example Up: Generalized Hermitian Eigenproblems Previous: Specifying an Eigenproblem Contents Index Related Eigenproblems 1. If and are Hermitian, choice of real numbers is not positive definite, but and is positive definite for some , one can solve the generalized Hermitian eigenproblem instead.

### A Basis for theTop Homology of a Generalized Partition Lattice

by Brian

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