DOI: 10.24193/subbmath.2017.0009
Determinantal inequalities for J -accretive dissipative matrices
Nat´ alia Bebiano and Jo˜ ao da Providˆ encia
Abstract. In this note we determine bounds for the determinant of the sum of twoJ-accretive dissipative matrices with prescribed spectra.
Mathematics Subject Classification (2010):46C20, 47A12.
Keywords: J-accretive dissipative matrix, J-selfadjoint matrix, indefinite inner norm.
1. Results
Consider the complexn-dimensional spaceCnendowed with the indefinite inner product
[x, y]J=y∗J x, x, y∈Cn, whereJ =Ir⊕ −In−r, and correspondingJ-norm
[x, x]J =|x1|2+. . .+|xr|2− |xr+1|2−. . .− |xn|2.
In the sequel we shall assume that 0< r < n, except where otherwise stated.
TheJ-adjoint ofA∈Cn×n is defined and denoted as [A#x, x] = [x, Ax]
or, equivalently,A#:=J A∗J, [4]. The matrixAis said to beJ-HermitianifA#=A, and isJ-positive definite(semi-definite) ifJ Ais positive definite (semi-definite). This kind of matrices appears on Quantum Physics and in Symplectic Geometry [10]. An arbitrary matrixA∈Cn×n may be uniquely written in the form
A= ReJA+iImJA, where
ReJA= (A+A#)/2, ImJA= (A−A#)/(2i)
areJ-Hermitian. This is the so-called J-Cartesian decomposition ofA.J-Hermitian matrices share properties with Hermitian matrices, but they also have important differences. For instance, they have real and complex eigenvalues, these occurring in
conjugate pairs. Nevertheless, the eigenvalues of aJ-positive matrix are all real, being rpositive andn−rnegative, according to theJ-norm of the associated eigenvectors being positive or negative. A matrixA is said to beJ-accretive(resp.J-dissipative) ifJReJA(resp.JImJA) is positive definite. If both matricesJReJAandJImJAare positive definite the matrix is said to beJ-accretive dissipative. We are interested in obtaining determinantal inequalities forJ-accretive dissipative matrices. Determinan- tal inequalities have deserved the attention of researchers, [2], [3], [5]-[9], [11].
Throughout, we shall be concerned with the set
DJ(A, C) ={det(A+V CV#) :V ∈ U(r, n−r)},
where A, C ∈ Cn×n are J-unitarily diagonalizable with prescribed eigenvalues and U(r, n−r) is the group ofJ-unitary transformations inCn(V isJ-unitaryifV V#= I), [12]. The so-calledJ-unitary group is connected, nevertheless it is not compact. As a consequence,DJ(A, C) is connected. This set is invariant under the transformation C → U CU# for every J-unitary matrix U, and, for short, DJ(A, C) is said to be J-unitarily invariant.
In the sequel we use the following notation. By Sn we denote the symmetric group of degreen, and we shall also consider
Snr={σ∈Sn:σ(j) =j, j=r+ 1, . . . , n}, (1.1) Sˆnr={σ∈Sn:σ(j) =j, j= 1, . . . , r}. (1.2) Let αj, γj ∈ C, j = 1, . . . , n denote the eigenvalues of A and C, respectively. The r!(n−r)! points
zσ=zξτ =
r
Y
j=1
(αj+γξ(j))
n
Y
j=r+1
(αj+γτ(j)), ξ∈Snr, τ ∈Sˆnr. (1.3)
belong toDJ(A, C).
The purpose of this note, which is in the continuation of [1], is to establish the following results.
Theorem 1.1. LetJ =Ir⊕−In−r, andAandCbeJ- positive matrices with prescribed real eigenvalues
α1≥. . .≥αr>0> αr+1≥. . .≥αn (1.4) and
γ1≥. . .≥γr>0> γr+1≥. . .≥γn, (1.5) respectively. Then
|det(A+iC)| ≥ (α21+γ12). . .(α2n+γn2)1/2 .
Corollary 1.2. LetJ =Ir⊕ −In−r, andB be aJ-accretive dissipative matrix. Assume that the eigenvalues ofReJB andImJB satisfy (1.4) and (1.5), respectively. Then,
|det(B)| ≥ (α21+γ12). . .(α2n+γn2)1/2 .
Example 1.3. In order to illustrate the necessity ofAandCto beJ-positive matrices in Theorem 1.1, letA= diag(α1, α2), C= diag(γ1, γ2), withα1=γ1= 1,α2= 3/2, γ2 = −2, and J = diag(1,−1). We find (α21+γ21)(α22+γ22) = 27/2. However, the minimum of|det(A+iV BV#|2, forV ranging over theJ-unitary group , is 49/4.
Theorem 1.4. LetJ =Ir⊕ −In−r, andAandCbeJ-unitary matrices with prescribed eigenvalues
α1, . . . , αr, αr+1, . . . , αn
and
γ1. . . , γr, γr+1, . . . .γn, respectively. Assume moreover that
=α1
2(1 +<α1) ≤. . .≤ =αr
2(1 +<αr)<0< =αr+1
2(1 +<αr+1) ≤. . .≤ =αn
2(1 +<αn) (1.6) and
=γ1
2(1− <γ1)≤. . .≤ =γr
2(1− <γr) <0< =γr+1
2(1− <γr+1) ≤. . .≤ =γn
2(1− <γn). (1.7) Then
DJ(A, C) = (α1+γ1). . .(αn+γn)[1,+∞[ .
We shall present the proofs of the above results in the next section.
2. Proofs
Lemma 2.1. Let g:U(r, n−r)→Rbe the real valued function defined by g(U) = det(I+A−10 U C0J U∗J A−10 U C0J U∗J),
whereA0= diag(α1, . . . , αn), C0= diag(γ1, . . . , γn)andαi, γj satisfy (1.4) and (1.5).
Then the set
{U ∈ U(r, n−r) :g(U)≤a}, where
a >
n
Y
j=1
1 + γ2j α2j
! ,
is compact.
Proof. Notice thatJ A0>0, J C0>0,so we may write g(U) = det(I+W W∗W W∗), where
W = (J A0)−1/2U(J C0)1/2.
The conditiong(U)≤aimplies thatW is bounded, and is satisfied if we require that W W∗ ≤κI, forκ >0 such that (1 +κ2)n≤a.Thus, also U is bounded. The result
follows by Heine-Borel Theorem.
Proof of Theorem 1.1
Under the hypothesis, A is nonsingular. Since the determinant is J-unitarily invariant andC isJ-unitarily diagonalizable, we may considerC= diag(γ1, . . . , γn).
We observe that
|det(A+iC)|2= det ((A+iC)(A−iC)) =
n
Y
i=1
αi
!2
det (I+iA−1C)(I−iA−1C)
Clearly,
det (I+iA−1C)(I−iA−1C)
= det(I+A−1CA−1C).
The set of values attained by|det(A+iC)|2is an unbounded connected subset of the positive real line. In order to prove the unboundedness, let us consider theJ-unitary matrixV obtained from the identity matrixI through the replacement of the entries (r, r), (r+ 1, r+ 1) by cosh u, and the replacement of the entries (r, r+ 1), (r+ 1, r) by sinh u,u∈R. We may assume thatA0= diag(α1, . . . , αn). A simple computation shows that
|det(A0+iV CV#)|2=
n
Y
j=1
(α2j+γj2)
−2(αr−αr+1)(γr−γr+1)(αr+1γr+αrγr+1)(sinh u)2 + (αr−αr+1)2(γr−γr+1)2(sinh u)4.
Thus, the set of values attained by|det(A0+iV CV#)|is given by [(α21+γ12)1/2. . .(α2n+γn2)1/2,+∞[.
As a consequence of Lemma 2.1, the set of values attained by|det(A+iC)|2is closed and a half-ray in the positive real line. So, there exist matrices A, C such that the endpoint of the half-ray is given by|det(A+iC)|2. Let us assume that the endpoint of this half-ray is attained at|det(A+iC)|2. We prove thatAcommutes withC. Indeed, for∈Rand an arbitraryJ-HermitianX, let us consider theJ-unitary matrix given as
eiX =i+iX−2
2X2+. . . . We obtain by some computations
f() := det(I+A−1e−iXCeiXA−1e−iXCeiX)
= det(I+A−1CA−1C−i(A−1[X, C]A−1C+A−1CA−1[X, C]) +O(2)
= det(I+A−1CA−1C)
×det I−i(I+A−1CA−1C)−1(A−1[X, C]A−1C+A−1CA−1[X, C])
+O(2)
= det(I+A−1CA−1C)
×exp −itr((I+A−1CA−1C)−1(A−1[X, C]A−1C+A−1CA−1[X, C]))
+O(2),
where [X, Y] = XY −Y X denotes the commutator of the matrices X and Y. The functionf() attains its minimum at det(I+A−1CA−1C), if
df d =0
= 0.
Then we must have
tr (I+A−1CA−1C)−1(A−1[X, C]A−1C+A−1CA−1[X, C])
= 0, for everyJ-HermitianX. That is
[C,(A−1C(I+A−1CA−1C)−1A−1+ (I+A−1CA−1C)−1A−1CA−1)] = 0, and so, performing some computations, we find
[C,(A−1C(I+A−1CA−1C)−1A−1C+ (I+A−1CA−1C)−1A−1CA−1C)]
= 2
C, A−1CA−1C I+A−1CA−1C
= 2
C, I− I
I+A−1CA−1C
=−2
C, I
I+A−1CA−1C
= 2I
I+ (A−1C)2
C, (A−1C)2 I
I+ (A−1C)2 = 0.
Thus
[C,(A−1C)2] = 0.
Assume thatC, which is in diagonal form, has distinct eigenvalues. Then (A−1C)2 is a diagonal matrix as well as ((J A)−1J C)2. Furthermore, ((J C)1/2(J A)−1(J C)1/2)2 is diagonal. Since (J C)1/2(J A)−1(J C)1/2 is positive definite, it is also diagonal, and so are (J A)−1J C andA−1C . Henceforth,Ais also a diagonal matrix and commutes withC. (IfChas multiple eigenvalues we can apply a perturbative technique and use a continuity argument).
Forσ∈Sn, such thatσ(1), . . . , σ(r)≤r,we have
(α21+γσ(1)2 ). . .(α2n+γσ(n)2 )≥(α21+γ21). . .(α2n+γn2).
Thus, the result follows.
In the proof of Theorem 1.4, the following lemma is used (cf. [1, Theorem 1.1]).
Lemma 2.2. Let B, D beJ-positive matrices with eigenvalues satisfying β1≥. . .≥βr>0> βr+1≥. . . > βn,
and
δ1≥. . .≥δr>0> δr+1≥. . . > δn. Then
DJ(B, D) ={(β1+δ1). . .(βn+δn) t:t≥1}.
Proof of Theorem 1.4
Since, by hypothesis, A, C, are J-unitary matrices, considering convenient M¨obius transformations, it follows that
B = i 2
A−I
A+I, D=−i 2
C+I
C−I (2.1)
areJ-Hermitian matrices. Since
B+D=−i(A+I)−1(C+A)(C−I)−1, we obtain
det(B+D) =in det(A+C) Qn
j=1(1 +αj)(1−γj). Assume that the eigenvalues ofB andD are
σ(B) ={β1, . . . , βn}, σ(D) ={δ1, . . . , δn},
respectively. From (2.1) we get, βj =− =αj
2(1 +<αj), δj=− =γj
2(1− <γj). From (1.6) and (1.7) we conclude that
β1≥. . .≥βr>0> βr+1≥. . . > βn, and
δ1≥. . .≥δr>0> δr+1≥. . . > δn,
so that the matricesB and DareJ-positive. From Lemma 2.2 it follows that DJ(B, D) = (β1+δ1). . .(βn+δn)[1,+∞[.
Thus,DJ(A, C) is a half-line with endpoint at (α1+γ1). . .(αn+γn), or, more precisely,
DJ(A, C) ={(α1+γ1). . .(αn+γn)t:t≥1}.
Acknowledgments.This work was partially supported by the Centro de Matem´atica da Universidade de Coimbra (CMUC), funded by the European Regional Develop- ment Fund through the program COMPETE and by the Portuguese Government through the FCT - Funda¸c˜ao para a Ciˆencia e a Tecnologia under the project PEst- C/MAT/UI0324/2011.
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Nat´alia Bebiano
CMUC, Departament of Mathematics Universidade de Coimbra
3001-454 Coimbra, Portugal e-mail:[email protected] Jo˜ao da Providˆencia Departamento de F´ısica Universidade de Coimbra 3001-454 Coimbra, Portugal e-mail:[email protected]
