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Generalized Symmetric Definite Eigenproblems

The generalized eigenvalue problems are defined as tex2html_wrap_inline1411 , or tex2html_wrap_inline1413 , or tex2html_wrap_inline1415 , where A and B are real symmetric or complex Hermitian and B is positive definite. Each of these problems can be reduced to a standard symmetric eigenvalue problem, using a Cholesky factorization of B as either tex2html_wrap_inline1425 or tex2html_wrap_inline1427 ( tex2html_wrap_inline1429 or tex2html_wrap_inline1431 in the Hermitian case).

With tex2html_wrap_inline1425 , we have


Hence the eigenvalues of tex2html_wrap_inline1411 are those of tex2html_wrap_inline1437 , where C is the symmetric matrix tex2html_wrap_inline1441 and tex2html_wrap_inline1443 . In the complex case C is Hermitian with tex2html_wrap_inline1447 and tex2html_wrap_inline1449 .

Table 1 summarizes how each of the three types of problem may be reduced to standard form tex2html_wrap_inline1437 , and how the eigenvectors z of the original problem may be recovered from the eigenvectors y of the reduced problem. The table applies to real problems; for complex problems, transposed matrices must be replaced by conjugate-transposes.

Table: Reduction of generalized symmetric definite eigenproblems to standard problems

Susan Blackford
Thu Jul 25 15:38:00 EDT 1996