An Extension of the Roots Separation Theorem |
| |
Authors: | Erxiong Jiang |
| |
Institution: | (1) Department of Mathematics, Shanghai University, Shanghai, PR, China |
| |
Abstract: | Let T
n
be an n×n unreduced symmetric tridiagonal matrix with eigenvalues 1< 2<![sdot](/content/kr46274q512x5558/xxlarge8901.gif) ![sdot](/content/kr46274q512x5558/xxlarge8901.gif) <
n
and W
k
is an (n–1)×(n–1) submatrix by deleting the kth row and the kth column from T
n
, k=1,2,...,n. Let 1![le](/content/kr46274q512x5558/xxlarge8804.gif) 2![le](/content/kr46274q512x5558/xxlarge8804.gif) ![sdot](/content/kr46274q512x5558/xxlarge8901.gif) ![sdot](/content/kr46274q512x5558/xxlarge8901.gif) ![sdot](/content/kr46274q512x5558/xxlarge8901.gif) ![le](/content/kr46274q512x5558/xxlarge8804.gif)
n–1 be the eigenvalues of W
k
. It is proved that if W
k
has no multiple eigenvalue, then 1< 1< 2< 2<![sdot](/content/kr46274q512x5558/xxlarge8901.gif) ![sdot](/content/kr46274q512x5558/xxlarge8901.gif) <
n–1<
n–1<
n
; otherwise if
i
=
i+1 is a multiple eigenvalue of W
k
, then the above relationship still holds except that the inequality
i
<
i+1<
i+1 is replaced by
i
=
i+1=
i+1. |
| |
Keywords: | eigenvalue problem symmetric tridiagonal matrix interlace theorem divide-and-conquer method |
本文献已被 SpringerLink 等数据库收录! |
|