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12 A. KOVACˇEC
[BdP97, determinantal analogue to Miranda-Thompson trace inequality]: It is shown first that in above notation the following are equivalent:
i. D(A, B) is a disk; ii.α1≥βn &β1≥αn;
iii. 0 ∈ R(A0, B0).
Theorem. If A,B ∈ Mn(R) and detA ≥ 0, detB ≥ 0, then
n
max det(A + UBV ) = (αi + βn−i+1).
U,V ∈SO(n)
i=1
Analogously the minimum minU,V ∈SO(n) det(A + UBV ) is determined but splits into four distinct cases defined by the parity of n and inequalities that hold between α1, β1, αn, βn.
[BPer98, determinantal analogue to von Neumann’s trace inequality and inequal- ities for Hadamard products]: Here in the first part Nat´alia and Cec´ılia Perdig˜ao continue the previous topic and define the additional set
D(A, B) = {det(A′ + B′) : A′ symmetric, B′ skew symmetric
with singular values of A, B, respectively}
Thanks to a classical result and a result of Youla, it is known how to range efficiently over the herein admitted A′, B′, respectively. For A, B ∈ Mn let
rB(A) = max{|z| : z ∈ D(A,B)} and RB(A) = max{|z| : z ∈ D(A,B)}.
In Section 2 the authors find for the maximum and the minimum of the modulus of det(A + U BV ) the following.
(5.1) and
(5.2)
n
max |det(A+UBV)|= (αi +βn−i+1)
min
U,V ∈U(n)
U,V ∈U(n)
|det(A + UBV )|=
i=1
n |αi−βn−i+1|
i=1
if αn>β1 or βn>α1, otherwise.
0
