Absolute and relative Weyl theorems for generalized eigenvalue problems (Q1044601)

The author considers a generalized eigenvalue problem \(Ax=\lambda Bx\) with \(A\) and \(B\) Hermitian and \(B\) positive definite. Suppose its ordered eigenvalues are \(\lambda_1\leq\cdots\leq\lambda_n\). Replacing \(A\) by \(\tilde{A}=A+\Delta A\) and \(B\) by \(\tilde{B}=B+\Delta B\) results in an ordered set of perturbed eigenvalues \(\tilde{\lambda}_1\leq\cdots\leq\tilde{\lambda}_n\). If \(\|\Delta B\|_2\) is small enough, an upper bound is given for \(|\lambda_i-\tilde{\lambda}_i|\), \(i=1,\dots,n\) which reduces in the case \(B=\tilde{B}=I_n\) to the classical Weyl bound. For the relative perturbations, \(\tilde{A}\) is given by \((I+\Delta A)^T A(I+\Delta A)\) and similarly \(\tilde{B}=(I+\Delta B)^T A(I+\Delta B)\). In that case an expression of the form \(|\lambda_i-\tilde{\lambda}_i|\leq C|\lambda_i|\) is derived where \(C\) depends on the numbers \(\epsilon(A)\) and \(\epsilon(B)\) with \(\epsilon(M)=\sqrt{\kappa_2(M)}\|\Delta M\|_2\) where \(\kappa_2\) is the usual condition number and \(M\in\{A,B\}\). Also an expression is given when \(B\) gets an absolute perturbation and \(A\) gets a relative one, in which case \(C\) will depend on \(\|\Delta B\|_2\) and \(\epsilon(A)\).