Let A and B be n-by-n matrices. Recall that the numerical radius of A is defined by wA sup | Ax, x ∈ C ∶ x ∈ , |x| 1. For the numerical radius of the matrix product AB, the following inequalities are well known. (1) wAB 4wAwB. (2) wAB 2wAwB if AB=BA. (3) wAB ||A||wB if AB=BA and A is normal. (4) wAB ||A||wB if AB=BA and A*B=BA*. (5) wAB wAwB if n=2 and AB=BA. In this project, we consider when these inequalities become equalities. We want to obtain necessary and sufficient conditions for these equalities to hold, respectively. For each inequality, we have given the conjecture for the equality to hold, we will prove these conjectures as the purpose of this project.
|Effective start/end date||1/08/15 → 31/07/16|
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