On an isolated eigenvalue of a matrix and the structure of the corresponding eigenvector

✍ Scribed by V.A. Pupkov

• Publisher: Elsevier Science
• Year: 1983
• Weight: 375 KB
• Volume: 23
• Category: Article
• ISSN: 0041-5553
• DOI: 10.1016/s0041-5553(83)80070-6

✦ Synopsis

The disposition of the eigenvalues of general matrices and of Hermitian and symmetric matrices with isolated Gershgorin discs and "almost" isolated Gershgorin discs is made more precise.

To improve the localization of the eigenvalues, information about the structure of the corresponding eigenvector is used.

A sufficient condition is given for a matrix to be non-degenerate.

We introduce the following notation: N={I, 2,...,n},N(i,,...,&) is a subset of N, obtained by discarding elements it, .... i~;A=llao[] is an n x n matrix, D(~,) is a diagonal matrix of order n with elements d,=~,, d~=J V]~N(i); B(~,)=D(~,)AD-'(~,); T,= max la~,l, P'=Z laJ, q,= ~, la~,l; j~N(i) J~N(1) l~N{i)