Page 18 - Textos de Matemática Vol. 44

8 A. KOVACˇEC
for nonsingular P used by Fiedler for his aforementioned result.
[BMdP87, OMC, n=3]: This paper gives a proof of the Oliveira-Marcus con- jecture for 3 × 3 matrices based on Birkhoff’s theorem for doubly stochastic matrices. It is known by work of Drury that the most natural tentative to extend this proof to the case n ≥ 4 fails.
[BdP86, bound for determinant of strictly dissipative matrices]: This is another early paper possibly inspired by OMC. Recall that a complex square matrix written in cartesian decomposition M = H + iK is strictly dissipative if K is positive definite.
Theorem. Assume α1 ≥ ... ≥ αn and β1 ≥ ... ≥ βn > 0 to be the eigenvalues of H and K, respectively, in a strictly dissipative matrix M = H + iK. Then there hold the bounds
nn
|αj +iβj|≤|detM|≤ |αj +iβn−j+1|, j=1 j=1
with left and right inequalities turning equalities iff M is a normal matrix with eigenvalues αj + iβj and αj + iβn−j+1, respectively.
This generalizes the result [B86] where both matrices had to be positive semidefinite; what singles out this paper among other papers of Nat´alia’s that deal with containment regions for determinants of the form det(A + B) is that it contains also bounds for the argument of det(H + iK).
Theorem. If the matrices H and K are positive definite, then we have
nπ − β−1αn−j+1 + β−1αj − arctan β−1αj ≤ arg det(H + iK)
2jjj jjj
≤ nπ − β−1αj + β−1αn−j+1 − arctan β−1αn−j+1. 2jjj
jjj
Proof. We sketch the proof of this latter theorem. It is easy to see that we
have
n det(H+iK)= (ηj +i)detK,
j=1
with ηj, j = 1,...,n, being the eigenvalues of HK−1. Thus
arg det(H + iK) = nπ − arctan ηj = nπ − tr(arctan(K−1/2HK−1/2))
j
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