A result complementary to Geršgorin's circle theorem
\textit{S. Geršgorin's} theorem [Izv. Akad. Nauk SSSR, Otd. Mat. Estest. Nauk, VII. Ser. No. 6, 749--754 (1931; Zbl 0003.00102)] about circles in the complex plane containing an eigenvalue of a complex matrix \(M\), though simple to prove, has given rise to a large area known as localization theory, on which \textit{R. S. Varga} has written a book [Geršgorin and his circles. Springer Series in Computational Mathematics 36. Berlin: Springer (2004; Zbl 1057.15023)]. In this paper the author proves a result which is complementary to Geršgorin's theorem in two ways. First, it concerns eigenvectors of \(M\) rather than eigenvalues, and secondly, it examines the consequences of the corresponding eigenvalue not lying in a Geršgorin circle. Though elementary, the result yields an economical way of determining where an eigenvector of a narrow banded matrix is negligible, and finds application in physics research on the ``localization'' of eigenvectors.
scientific article; zbMATH DE number 5561606