Page 130 - Textos de Matemática Vol. 39

120 JOA˜O FILIPE QUEIRO´
Another comment is that in general we cannot expect the vertices of the realizable set to be produced with diagonal matrices. This is already the case in the n = 3 example above. The next illustration shows the polygon E(α, β) in R3 for the same triples α and β:
7 6.5 6 5.5 5 4.5 4 3.5
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6
10 11
7
8
9
12
10 13
We have marked with a black dot the six points in E(α, β) produced with diagonal A and B. There are four vertices not obtained with diagonal matrices: (10, 10, 4), (9, 9, 6), (10, 7, 7), (12, 6, 6).
That is why in general the partial spectrum problem does not follow trivially from the full spectrum case: we wish to describe the projection of E(α, β) onto the coordinate s-plane spanned by the k1, . . . , ks canonical vectors, but we don’t know the vertices of E(α, β).
5. The case s=n−1
Here we are interested in the possible (n − 1)-tuples
γ = (γ1,...,γh−1,γh+1,...,γn)
occurring as part of the spectra of sums A + B, A and B Hermitian with the given spectra α and β.
This is the simplest case of all, because the missing eigenvalue is deter- mined from the trace condition.