Norm inequalities for cartesian decompositions Original Research Article

Let the Cartesian decomposition of a complexn × n matrixT beT = A + iB withA, B Hermitian. Letαj andβj be the eigenvalues ofA andB respectively ordered so thatα1greater-or-equal, slanted … greater-or-equal, slanted αnandβ1greater-or-equal, slanted … greater-or-equal, slanted βn. We prove thatshort paralleldiag(α1+iβ1,…αn+iβn)short parallelless-than-or-equals, slant2short parallelTshort parallel 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).