Optimal bounds for the spectral variation of two regular matrix pairs (Q855582)

For \(n\times n\) matrices \(A\), \(B\) over \(\mathbb C\), call \(Z=(A,B)\) a regular matrix pair if \(\det(A-\lambda B)\neq0\) for \(\lambda\in\mathbb C\), and for \(\alpha\), \(\beta\in\mathbb C\), call \((\alpha, \beta)\) an eigenvalue of \(Z\) if \(\det(\beta A-\alpha B)=0\). Let \(Z=(A,B)\) and \(W=(C,D)\) are two regular pairs with eigenvalues \((\alpha_j,\beta_j)\) and \((\gamma_i,\delta_i)\), respectively. The spectral variation of \(W\) with respect to Z is defined to be \[ S_Z(W)=\max_{i}\min_{j}\frac{| \alpha_j\delta_i-\beta_j\gamma_i| }{{\sqrt{| \alpha_j| ^2+| \beta_j| ^2}}{\sqrt{| \gamma_i| ^2+| \delta_i| ^2}}}. \] Results by L. Elsner for upper bounds of spectral variation are generalized. One of them states that \[ S_Z(W)\leq \frac{1}{D(Z)}\| Z\| ^{1-1/n}\| Z-W\| ^{1/n}, \] in which \[ D(Z)=(\max_{| \alpha| ^2+| \beta| ^2=1}| \det(\beta A-\alpha B)| )^{1/n}, \] and the spectral norm is used.