On the Fan and Lidskii Majorisations of Positive Semidefinite Matrices
نویسنده
چکیده
Let A,B be two positive definite matrices. The Fan-Lidskii majorisations can be subsumed as the symmetric norm rearrangement inequalities ‖A↓ +B↑‖ ‖A+B‖ ‖A↓ +B↓‖ where the up/down arrows on A,B mean the diagonal matrices with the same eigenvalues in decreasing/increasing order down to the diagonal. We refine these relations with sums of type A +UBU∗ and A +VBV∗ for two unitary matrices U,V associated in a quite natural way to A and B . Stronger results than majorisation are given by using simple averages in the unitary orbits of A+B , A+UBU∗ and A+VBV ∗ . Proofs rely on an orbital decomposition for positive block-matrices. Mathematics subject classification (2010): 15A60, 47A30, 15A42.
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