A note on the Hoffman-Wielandt theorem for generalized eigenvalue problems (Q1036910)

Let \(A\) and \(B\) be complex \(n\times n\) matrices with eigenvalues \(\alpha_1,\dots,\alpha_n\) and \(\beta_1,\dots,\beta_n\) respectively. \textit{A.~J.~Hoffman} and \textit{H.~W.~Wielandt} [Duke Math. J.~20, 37--39 (1953; Zbl 0051.00903)] proved that if \(A\) and \(B\) are normal, then \[ \min_{\sigma\in S_n}\sum_{i=1}^n|\alpha_i-\beta_{\sigma(i)}|^2\leq \|A-B\|_F^2, \] where \(\|.\|_F\) denotes the Frobenius norm. The matrix pair \(\{A,B\}\) is regular if \(\det{(A+\lambda B)}\) is not identically zero. This pair is diagonalizable if it is regular and there exist invertible matrices \(P\) and \(Q\) such that \(A=P\Lambda Q\), \(B=P\Omega Q\), and \(\Lambda\) and \(\Omega\) are diagonal matrices. Furthermore, \(\{A,B\}\) is normal if it is diagonalizable and \(Q\) is unitary. \textit{J.~G.~Sun} [Math. Numer. Sinica~4, 23--29 (1982; Zbl 0537.15005)] stated the Hoffman-Wielandt theorem for normal pairs. \textit{R.-C.~Li} [Math. Comp.~72, 715--728 (2003; Zbl 1047.15008)] extended it to diagonalizable pairs. The present authors give another extension to diagonalizable pairs. In these studies, the distance between two points of the complex plane is defined by the chordal metric \[ \rho((\alpha,\beta),(\gamma,\delta))=\frac{|\alpha\delta-\beta\gamma|} {\sqrt{|\alpha|^2+|\beta|^2}\sqrt{|\gamma|^2+|\delta|^2}},\qquad (\alpha,\beta),(\gamma,\delta)\neq (0,0). \] The distance between two regular matrix pairs \(\{A,B\}\) and \(\{C,D\}\) is defined by \[ d_F(Z,W)=\frac{1}{\sqrt{2}}\|P_{Z^*}-P_{W^*}\|_F, \] where \(P_X\) denotes the orthogonal projector onto the column space of a matrix~\(X\), \(Z=(A,B)\) and \(W=(C,D)\).