On spectral variation of two-parameter matrix eigenvalue problem (Q2834187)

Let \(Z_{j}\), \(A_{j1}\) and \(A_{j2}\) be \(n_{j}\times n_{j}\) complex matrices for \(j=1\) and \(2\). Consider the pair \(\Lambda\) of generalized pencils \(Z_{1}-\lambda_{1}A_{11}-\lambda_{2}A_{12}\) and \(Z_{2}-\lambda_{1} A_{21}-\lambda_{2}A_{22}\). Then the pair \((\lambda_{1},\lambda_{2})\) of complex numbers is called an eigenvalue of \(\Lambda\) if both of these generalized pencils are singular. Now consider a second pair \(\tilde{\Lambda}\) of generalized pencils \(\tilde{Z}_{1}-\lambda_{1}\tilde{A}_{11}-\lambda_{2}\tilde{A}_{12}\) and \(\tilde{Z}_{2}-\lambda_{1}\tilde{A}_{21}-\lambda_{2}\tilde{A}_{22}\) of the same dimensions as before. We shall think of \(\tilde{\Lambda}\) as a perturbation of \(\Lambda\). The spectral variation of \(\tilde{\Lambda}\) with respect to \(\Lambda\) is defined to be the pair \((sv_{\Lambda}^{(1)}(\tilde{\Lambda}),sv_{\Lambda}^{(2)}(\tilde{\Lambda}))\) where \(sv_{\Lambda}^{(j)}(\tilde{\Lambda}):=\sup_{s}\inf_{t}\left| s-t\right| \) where \(s\) runs over the \(j\)th component of the eigenvalues of \(\tilde{\Lambda}\) and \(t\) runs over the \(j\)th component of the eigenvalues of \(\Lambda\). Next for any square matrix \(A\) we define \(r_{s}(A)\) to be the spectral radius, \(\left\| A\right\| \) to be the spectral norm and \(N_{p}(A)\) (\(1\leq p<\infty\)) to be the Schatten-von Neumann norm \(\left(\operatorname{Trace}(AA^{\ast}\right) ^{p/2})^{1/p}\). Finally define \(K_{0}:=A_{11}\otimes A_{22}-A_{12}\otimes A_{21}\), \(K_{1}:=Z_{1}\otimes A_{22}-A_{12}\otimes Z_{2}\) and \(K_{2}:=A_{11}\otimes Z_{2}-Z_{1}\otimes A_{21}\) with analogous definitions for \(\tilde{K}_{0}\), \(\tilde{K}_{1}\) and \(\tilde{K}_{2}\), and set \(q(K_{\ell})=\left\| K_{\ell}-\tilde{K}_{\ell}\right\| \) for \(\ell=1,2\) and \(3\). Then the main theorem is as follows. If \(K_{0}\) and \(\tilde{K}_{0}\) are both invertible and \(p\geq2\) then NEWLINE\[NEWLINEsv_{\Lambda}^{(j)}(\tilde{\Lambda})^{n_{1}n_{2}}\leq c_{j}\left\{N_{p} (K_{j})+r_{s}(\tilde{K}_{0}^{-1}\tilde{K}_{j})N_{p}(K_{0})^{n_{1}n_{2} -1}\right\} \text{ for }j=1\text{ and }2 NEWLINE\]NEWLINE where NEWLINE\[NEWLINE c_{j}:=\frac{q(K_{j})+q(K_{0})r_{s}(\tilde{K}_{0}^{-1}\tilde{K}_{j})}{\left| \det(K_{0})\right| (n_{1}n_{2}-1)^{(n_{1}n_{2}-1)/p}}\text{.} NEWLINE\]NEWLINE A similar inequality holds when the norm \(N_{p}\) is replaced by the spectral norm.