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# Is there any generalizeda generalization of the Sylvester inertia law theoremInertia Law for the symmetric generalziedgeneralized eigenvalue problem?

I know that in order to solve symmetric eigenvalue problem $$Ax = \lambda x$$, we can use the Sylvester inertia lawInertia Law, that is the number of eigenvalues of A$$A$$ less than a$$a$$ equals the number of negative entries of $$D$$ where diagonal matrix D$$D$$ comes from ldlthe LDL factorization of $$A-aI = LDL^{T}$$. Then, by bisection method, we can find all or some eigenvalues as desired. Now I do notwish to know isif there any generalizedexists a generalization of the Sylvester Inertia lawLaw for thesymmetric generalized eigenvalue problems, that is solving $$Ax= \lambda Bx$$, where A$$A$$ and B$$B$$ are symmetric matrices. Thanks.

# Is there any generalized Sylvester inertia law theorem for symmetric generalzied eigenvalue problem?

I know that in order to solve symmetric eigenvalue problem $$Ax = \lambda x$$, we can use Sylvester inertia law, that is the number of eigenvalues of A less than a equals the number of negative entries of $$D$$ where diagonal matrix D comes from ldl factorization of $$A-aI = LDL^{T}$$. Then by bisection method, we can find all or some eigenvalues as desired. Now I do not know is there any generalized Sylvester Inertia law for the generalized eigenvalue problems, that is solving $$Ax= \lambda Bx$$, where A and B are symmetric matrices. Thanks.

# Is there a generalization of the Sylvester Inertia Law for the symmetric generalized eigenvalue problem?

I know that in order to solve symmetric eigenvalue problem $$Ax = \lambda x$$, we can use the Sylvester Inertia Law, that is the number of eigenvalues of $$A$$ less than $$a$$ equals the number of negative entries of $$D$$ where diagonal matrix $$D$$ comes from the LDL factorization of $$A-aI = LDL^{T}$$. Then, by bisection method, we can find all or some eigenvalues as desired. I wish to know if there exists a generalization of the Sylvester Inertia Law for symmetric generalized eigenvalue problems, that is solving $$Ax= \lambda Bx$$, where $$A$$ and $$B$$ are symmetric matrices. Thanks.

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# Is there any generalized Sylvester inertia law theorem for symmetric generalzied eigenvalue problem?

I know that in order to solve symmetric eigenvalue problem $$Ax = \lambda x$$, we can use Sylvester inertia law, that is the number of eigenvalues of A less than a equals the number of negative entries of $$D$$ where diagonal matrix D comes from ldl factorization of $$A-aI = LDL^{T}$$. Then by bisection method, we can find all or some eigenvalues as desired. Now I do not know is there any generalized Sylvester Inertia law for the generalized eigenvalue problems, that is solving $$Ax= \lambda Bx$$, where A and B are symmetric matrices. Thanks.