Page 23 - Textos de Matemática Vol. 44

NATA´LIA BEBIANO’S LEGACY 13
This result was actually already achieved in the previous paper, but in the present one a full characterization of the case of equality in (5.1), resp. (5.2) is given.
Since D(A, B) ⊆ D(A, B), it is clear that rB (A) ≤ RB (A). The next result characterizes the cases of equality.
Theorem. Let B ∈ Mn(C). Then the following are equivalent. i. B is symmetric.
ii. rB(A) = RB(A) for all A.
iii. There exists a symmetric A with s.v. α1 > ... > αn > 0 so that rB(A) =
RB (A).
A similar theorem is presented for skew symmetric B.
The paper presents further a number of determinantal inequalities for Hadamard products of positive semidefinite matrices. Collect for positive semi- definite A its eigenvalues in the decreasing n-tuple [λ(A)] = λ1(A) ≥ ... ≥ λn(A).WriteA≽Bifλ(A)≽λ(B),andA≽log BiflogA≽logB.Let detk(A) be the product of the k largest eigenvalues of A and recall the def- inition A#B = A1/2(A−1/2BA−1/2)A1/2 of the geometric mean of positive definite matrices A,B. The paper has a number of results of the following type.
Theorem. Let A, B be positive definite and let p ∈ Z≥1 . Then
detk(I + (Ap ◦ Bp)−1) ≤ detk(I + (A#B)−2p) ≤ detk(I + (AB)−p).
[BPL04, matrix inequalities in statistical and quantum mechanics]: The paper has an extended introduction to some of the matrix trace inequalities in sta- tistical mechanics. For hermitian matrices H1,H2 the inequality
tr(eH1 +H2 ) ≤ tr(eH1 eH2 ),
found in the sixties independently by Golden, Thompson, and Symanzik was used to give upper bounds for the partition function tr(e−H/θ). Here H = H1 + H2 can be thought in quantum mechanics as the sum of kinetic and potential energy; the case of classical physics corresponds to commuting H1,H2 and yields equality. At the other hand, tools like the Lie-Trotter formula according to which one has
e−H/θ = lim (e−H1/nθe−H2/nθ)n, n→∞
and the fact that if A, B are positive definite, then tr(AB)2p+1 ≤ tr(A2B2)2p