TY - JOUR

T1 - Relative perturbation bounds for eigenvalues of symmetric positive definite diagonally dominant matrices

AU - Ye, Qiang

PY - 2009

Y1 - 2009

N2 - For a symmetric positive semidefinite diagonally dominant matrix, if its off-diagonal entries and its diagonally dominant parts for all rows (which are defined for a row as the diagonal entry subtracted by the sum of absolute values of off-diagonal entries in that row) are known to a certain relative accuracy, we show that its eigenvalues are known to the same relative accuracy. Specifically, we prove that if such a matrix is perturbed in a way that each off-diagonal entry and each diagonally dominant part have relative errors bounded by some ε, then all its eigenvalues have relative errors bounded by ε. The result is extended to the generalized eigenvalue problem.

AB - For a symmetric positive semidefinite diagonally dominant matrix, if its off-diagonal entries and its diagonally dominant parts for all rows (which are defined for a row as the diagonal entry subtracted by the sum of absolute values of off-diagonal entries in that row) are known to a certain relative accuracy, we show that its eigenvalues are known to the same relative accuracy. Specifically, we prove that if such a matrix is perturbed in a way that each off-diagonal entry and each diagonally dominant part have relative errors bounded by some ε, then all its eigenvalues have relative errors bounded by ε. The result is extended to the generalized eigenvalue problem.

KW - Diagonal dominant matrix

KW - Eigenvalues

KW - Relative perturbation

UR - http://www.scopus.com/inward/record.url?scp=73649111754&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=73649111754&partnerID=8YFLogxK

U2 - 10.1137/060676349

DO - 10.1137/060676349

M3 - Article

AN - SCOPUS:73649111754

SN - 0895-4798

VL - 31

SP - 11

EP - 17

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

IS - 1

ER -