TY - JOUR
T1 - Norm inequalities for cartesian decompositions
AU - Zhan, Xingzhi
PY - 1999/1/1
Y1 - 1999/1/1
N2 - Let the Cartesian decomposition of a complex n × n matrix T be T = A + iB with A, B Hermitian. Let αj and βj be the eigenvalues of A and B respectively ordered so that |α1| ≥ ⋯ ≥ |αn| and |β1| ≥ ⋯ ≥ |βn|. We prove that ∥diag(α1 + iβ1,⋯, αn + iβn)∥ ≤ √2∥Τ∥ for every unitarily invariant norm this settles affirmatively a conjecture of Ando and Bhatia (T. Ando, R. Bhatia, Eigenvalue inequalities associated with the cartesian decomposition, Linear and Multilinear Algebra 22 (1987) 133).
AB - Let the Cartesian decomposition of a complex n × n matrix T be T = A + iB with A, B Hermitian. Let αj and βj be the eigenvalues of A and B respectively ordered so that |α1| ≥ ⋯ ≥ |αn| and |β1| ≥ ⋯ ≥ |βn|. We prove that ∥diag(α1 + iβ1,⋯, αn + iβn)∥ ≤ √2∥Τ∥ for every unitarily invariant norm this settles affirmatively a conjecture of Ando and Bhatia (T. Ando, R. Bhatia, Eigenvalue inequalities associated with the cartesian decomposition, Linear and Multilinear Algebra 22 (1987) 133).
KW - Cartesian decomposition
KW - Eigenvalue
KW - Singular value
KW - Unitarily invariant norm
UR - https://www.scopus.com/pages/publications/0040219794
U2 - 10.1016/S0024-3795(98)10174-X
DO - 10.1016/S0024-3795(98)10174-X
M3 - 文章
AN - SCOPUS:0040219794
SN - 0024-3795
VL - 286
SP - 297
EP - 301
JO - Linear Algebra and Its Applications
JF - Linear Algebra and Its Applications
IS - 1-3
ER -