TY - JOUR
T1 - Combined perturbation bounds
T2 - II. Polar decompositions
AU - Li, Wen
AU - Sun, Wei-Wei
PY - 2007/9
Y1 - 2007/9
N2 - In this paper, we study the perturbation bounds for the polar decomposition A = QH where Q is unitary and H is Hermitian. The optimal (asymptotic) bounds obtained in previous works for the unitary factor, the Hermitian factor and singular values of A are σ r 2 ΔQ F 2 Δ F 2 , 1/2ΔH F 2 ΔA F 2 and Δ∑ F 2 ΔA F 2 , respectively, where ∑ = diag(σ 1, σ 2, σ r , 0 ) is the singular value matrix of A and σ r denotes the smallest nonzero singular value. Here we present some new combined (asymptotic) perturbation bounds σ r 2 ΔQ F 2 +1/2 ΔH F 2 ΔA F 2 and σ r 2 Delta;Q F 2 + Δ∑ F 2 ΔA F 2 which are optimal for each factor. Some corresponding absolute perturbation bounds are also given. © 2007 Science in China Press.
AB - In this paper, we study the perturbation bounds for the polar decomposition A = QH where Q is unitary and H is Hermitian. The optimal (asymptotic) bounds obtained in previous works for the unitary factor, the Hermitian factor and singular values of A are σ r 2 ΔQ F 2 Δ F 2 , 1/2ΔH F 2 ΔA F 2 and Δ∑ F 2 ΔA F 2 , respectively, where ∑ = diag(σ 1, σ 2, σ r , 0 ) is the singular value matrix of A and σ r denotes the smallest nonzero singular value. Here we present some new combined (asymptotic) perturbation bounds σ r 2 ΔQ F 2 +1/2 ΔH F 2 ΔA F 2 and σ r 2 Delta;Q F 2 + Δ∑ F 2 ΔA F 2 which are optimal for each factor. Some corresponding absolute perturbation bounds are also given. © 2007 Science in China Press.
KW - Perturbation
KW - Polar decomposition
KW - Singular value
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U2 - 10.1007/s11425-007-0099-z
DO - 10.1007/s11425-007-0099-z
M3 - 22_Publication in policy or professional journal
VL - 50
SP - 1339
EP - 1346
JO - Science in China, Series A: Mathematics, Physics, Astronomy
JF - Science in China, Series A: Mathematics, Physics, Astronomy
SN - 1006-9283
IS - 9
ER -