Page 113 - Matrices theory and applications
P. 113
5. Nonnegative Matrices
96
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20. Let a ∈ IR be given, a =(a 1 ,... ,a n ).
(a) Show that
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C(a):= {b ∈ IR | b a}
is a convex compact set. Characterize its extremal points.
(b) Show that
Y (a):= {M ∈ Sym (IR) | Sp M a}
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is a convex compact set. Characterize its extremal points.
(c) Deduce that Y (a) is the closed convex hull (actually the convex
hull) of the set
X(a):= {M ∈ Sym (IR) | Sp M = a}.
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(d) Set α = s n (a)/n and a := (α,... ,α). Show that a ∈ C(a), and
that b ∈ C(a)=⇒ b ≺ a .
(e) Characterize the set
{M ∈ Sym (IR) | Sp M ≺ a }.
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