On a theorem of Feingold
Let A be an \(n\times n\) matrix and let X be a nonsingular matrix such that \(X^{-1}AX\) is a block diagonal matrix \(diag(A_ 1,...,A_ s)\). If \(Y=(\bar X^ T)^{-1}\), then the \(A_ i\) give rise to conformal partitions \(X=(X_ 1|...| X_ s)\) and \(Y=(Y_ 1|...| Y_ s)\). The columns of \(X_ i\) \((Y_ i)\) form a basis of the right (left) hand invariant subspace associated with the spectrum of \(A_ i\). Let \(P_ i=X_ i\bar Y^ T_ i\) be the projector associated with the i-th left hand and right hand subspaces, let \(\eta =0\) and let E be an \(n\times n\) matrix with \(\| E\| =1\) in some norm. The main result of this paper reads as follows: If one of each of the pairs \(X_ i\) and \(Y_ i\) is an orthogonal basis, then all eigenvalues of \(A+\eta E\) lie in the union of the domains \(D_ i=\{z\in {\mathbb{C}}: \| (A_ i- zI)^{-1}\|^{-1}\leq \eta \sum^{s}_{i=1}\| P_ i\|_ 2\},\) where I denotes the identity matrix and \(\| \cdot \|_ 2\) the Euclidean norm. This extends an unpublished theorem due to D. G. Feingold, which itself is a simple generalization of the Bauer-Fike theorem. The bulk of the paper under review contains a careful discussion of the sources of weakness of the main result. [Reviewer's remark: in reference [1], replace the journal data by ''Numer. Math. 2, 137-141 (1960)''.]
scientific article; zbMATH DE number 3998019