Inclusion regions for matrix eigenvalues
The authors derive inclusion regions that contain at least \(j\) eigenvalues of a complex matrix \(A\). When \(A\) is Hermitian, these reduce to the regions discussed by \textit{N. Lehmann} [Zur Verwendung optimaler Eigenwerteingrenzungen bei der Lösung symmetrischer Matrixenaufgaben, Numer. Math. 8, 42--55 (1966; Zbl 0143.37505) and Berechnung von Eigenwertschrauben bei linearen Problemen, Arch. Math. 2, 139--147 (1949)]. Let \(A\) be \(n\times n\), let \(S\) be an \(n\times m\) matrix with orthonormal columns, let \(\tau_1\leq\tau_2\leq\cdots\leq \tau_m\) be the singular values of \((A-\rho I)S\), and let \(\rho\) be a complex number. If \(A\) is diagonalizable and \(A= X\Delta X^{-1}\) is an eigenvalue decomposition, then for each \(j\), \(1\leq j\leq m\), at least \(j\) eigenvalues of \(A\) lie in the disk \(\{z:| z-\rho|\leq \text{cond}(X)\tau_{-j}\}\). If \(A\) is not normal, at least \(j\) eigenvalues of \(A\) lie in the disk \(\{z:| z-\rho|\leq \tau_{-j}+\delta(A)\}\). Here \(\text{cond}(X)= \| X\|\,\| X^{-1}\|\) where \(\| X\|\) denotes the 2-norm, and \(\delta(A)\) denotes the departure from normality of \(A\). Examples show these disks are optimal for normal matrices, but if \(A\) is normal and the range of \(S\) is far from an invariant subspace of \(A\), or if \(A\) is not normal, these inclusion regions can be arbitrarily pessimistic. The authors also derive inclusion regions that are exteriors of disks and annular exclusion regions, when \(S\) has orthonormal columns. When those columns are not orthonormal, they derive inclusion regions involving generalized singular values of matrix pairs.
scientific article; zbMATH DE number 1864801
scientific article