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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.