and V has the form (1.2).
A Appendix: admissible quadruples
LetS0 andαben×n matrices, and letϑ1 and ϑ2 be matrices of sizesn×m1
and n×m2, respectively. The quadruple {α, S0, ϑ1, ϑ2} is called admissible (see the beginning of Section 2) if
S0 >0 and αS0−S0α∗ =iΛΛ∗, where Λ :=
ϑ1 ϑ2
. (A.1)
By definition the transfer function associated with the quadruple Σ ={α, S0, ϑ1, ϑ2}
is the matrix function WΣ given by (see (2.10) and (3.15)):
WΣ(z) =Im+iΛ∗S0−1(zIn−α)−1Λ, where m:=m1+m2. (A.2) With Σ we associate two other rational matrix functions:
ϕ1,Σ(z) =iϑ∗2S0−1(zIn−β1)−1ϑ1, whereβ1 :=α−iϑ1ϑ∗1S0−1; (A.3) ϕ2,Σ(z) =−iϑ∗1S0−1(zIn+β2)−1ϑ2, where β2 :=α−iϑ2ϑ∗2S0−1. (A.4) If Σ = {α, S0, ϑ1, ϑ2} is an admissible quadruple, then the same holds true for the associate quadruple
Σ# :={α, S0, ϑ2, ϑ1} (A.5) and a direct computation shows that
ϕ1,Σ#(−z) =ϕ2,Σ(z). (A.6)
Admissible quadruples are closely related to symmetric S-nodes (see [44, Section 2.2]). Indeed, if Σ ={α, S0, ϑ1, ϑ2}is an admissible quadruple, then the matrices
A:=α, S:=S0, Π1 := Λ, Γ1 :=S0−1Λ,
form a symmetric S-node (withJ =In) as defined in Section 2.2 of [44]. Using this connection, it is readily seen that item (iii) in the following lemma is a special case of Theorem 2.2.1 in [44] (see also [42], Theorem 17.1 in [7] or Corollary 1.15 in [41]).
Lemma A.1. Let Σ ={α, S0, ϑ1, ϑ2} be an admissible quadruple. Then (i) σ(α)⊂C+;
(ii) if, in addition, the pair {α,Λ} is controllable, then σ(α)⊂C+, and in that case S0 is uniquely determined by α and Λ. More precisely, S0 is given by
S0 = 1 2π
Z ∞
−∞
(λIn−α)−1ΛΛ∗(λIn−α∗)−1dλ. (A.7) (iii) WΣ is a rational Schur class function and its values on the real line are
unitary matrices. Moreover, we have Im−WΣ(z)WΣ(ζ)∗ =
=i(z−ζ)Λ∗S0−1(zIn−α)−1S0(ζIn−α)−1S0−1Λ, (A.8) Im−WΣ(z)∗WΣ(z) =i(z−z)Λ¯ ∗(¯zIn−α∗)−1S0−1(zIn−α)−1Λ. (A.9) Proof. In remains to prove items (i) and (ii). Since S0 is positive definite, the second part of (A.1) implies that
S0−1/2αS01/2
−
S0−1/2αS0−1/2∗
=iS0−1/2ΛΛ∗S0−1/2, S0−1/2ΛΛ∗S0−1/2 ≥0.
It follows that the numerical range of the matrix S0−1/2αS01/2 is a subset of C+, and henceσ(α) = σ(S0−1/2αS01/2)⊂C+. This proves item (i).
The first statement in item (ii) is a straightforward application of the classical Chen-Wimmer inertia theorem (see Corollary 1 in [33, Section 13.1]).
For (A.7) see [26] or Theorem I.4.1 in [21] (see, also (3.10)).
Lemma A.2. Let Σ ={α, S0, ϑ1, ϑ2} be an admissible quadruple, and parti-tion its transfer funcparti-tion WΣ as a 2×2 block matrix function,
WΣ(z) =
"
a(z) b(z) c(z) d(z)
#
, where a(z) has size m1×m1. (A.10) Then the associated functions ϕ1,Σ and ϕ2,Σ are given by
ϕ1,Σ(z) =c(z)a(z)−1 and ϕ2,Σ(z) =b(−z)d(−z)−1. (A.11) Proof. From (A.2) and Λ =
ϑ1 ϑ2
it follows that
a(z) =Im1 +iϑ∗1S0−1(zIn−α)−1ϑ1, c(z) =iϑ∗2S0−1(zIn−α)−1ϑ1. Since a is a proper rational matrix function whose value at infinity is the m1×m1 identity matrix, a classical inversion theorem (see, e.g., Theorem 2.2 in [6]) tells us that
a(z)−1 =Im1 −iϑ∗1S0−1(zIn−α×)−1ϑ1, where α× =α−iϑ1ϑ∗1S0−1 =β1. Hence, again using Theorem 2.2 in [6] and taking into account (A.3), we know that
c(z)a(z)−1 =iϑ∗2S0−1(zIn−α)−1ϑ1a(z)−1 =iϑ∗2S0−1(zIn−α×)−1ϑ1
=iϑ∗2S0−1(zIn−β1)−1ϑ1 =ϕ1,Σ(z), which proves the first part of (A.11).
To prove the second identity in (A.11) we use the associate admissible quadruple Σ# ={α, S0, ϑ2, ϑ1} and equality (A.6). Note that
ϑ2 ϑ1
= ϑ1 ϑ2
U, where U =
0 Im1
Im2 0
. (A.12)
Since U is unitary, it is clear from (A.12) and (A.2) that WΣ# = U∗WΣU. But then, using the special form of U, we obtain
WΣ# =U∗WΣU =
"
d(z) c(z) b(z) a(z)
#
. (A.13)
Applying the result of the first part of the proof to Σ# in place of Σ we see thatϕ1,Σ# =b(z)d(z)−1. But then (A.6) yields the second identity in (A.11).
Theorem A.3. Let ϕ be an m2 ×m1 rational matrix function. Then ϕ coincides with some function ϕ1,Σ associated with an admissible quadrupleΣ if and only if ϕ is strictly proper. In that case, the corresponding Σ can be obtained explicitly by using the following procedure.
Step 1. Let n be the McMillan degree of ϕ, and construct a minimal real-ization of ϕ:
ϕ(z) =iθ∗2(zIn−γ)−1θ1. (A.14) Step 2. ChooseX to be the unique positive definite solution of the algebraic
Riccati equation
γX−Xγ∗−iXθ2θ2∗X+iθ1θ∗1 = 0. (A.15) Step 3. Put
S0 =In, ϑ1 =X−1/2θ1, ϑ2 =X1/2θ2, (A.16) α=X−1/2γX1/2+iϑ1ϑ∗1. (A.17) With these choices the quadruple Σ = {α, S0, ϑ1, ϑ2} is admissible and ϕ coincides with its first associate function ϕ1,Σ. Moreover, in this case the pairs {α, ϑ1} and {α, ϑ2} are controllable.
Proof. First note that the minimality of the realization (A.14) is equivalent to the requirement that simultaneously the pair{θ2∗, γ}is observable and the pair {γ, θ1} is controllable. But then we can apply Proposition 2.2 in [23] to show that the Riccati equation (A.15) has a positive definite solution X, a result which has its roots in Kalman’s theory of mathematical systems [28]
(see also [32, pp. 358 and 369], where the uniqueness ofX is shown as well).
The remaining part of the proof is split into two parts.
Part 1. Using the definition of α in (A.17), the unique positive definite solution X of the Riccati equation (A.15), and the definitions of ϑ1 and ϑ2
in (A.16) we see that
α−α∗ =X−1/2γX1/2−X1/2γ∗X−1/2+ 2iϑ1ϑ∗1
=iX1/2θ2θ2∗X1/2−iX−1/2θ1θ∗1X−1/2+ 2iϑ1ϑ∗1
=iϑ2ϑ∗2−iϑ1ϑ∗1+ 2iϑ1ϑ∗1 =iϑ2ϑ∗2+iϑ1ϑ∗1.
Since S0 = In by definition, we conclude that Σ = {α, S0, ϑ1, ϑ2} is an admissible quadruple. Moreover, the associate function ϕ1,Σ is equal to ϕ.
Indeed, in this case using (A.17), we have
β1 =α−iϑ1ϑ∗1 =X−1/2γX1/2. (A.18) Thus X1/2β1X−1/2 =γ, and hence
ϕ1,Σ(z) =iϑ∗2S0−1(zIn−β1)−1ϑ1
=iθ∗2X1/2(zIn−β1)−1X−1/2θ1
=iθ∗2(zIn−X1/2β1X−1/2)−1θ1
=iθ∗2(zIn−γ)−1θ1 =ϕ.
Part 2. In this part we prove that the pairs {α, ϑ1}and{α, ϑ2}are control-lable. Since the realization (A.14) is minimal andϕ =ϕ1,Σ, the same is valid for the realization (A.3), and hence the pairs{β1, ϑ1}and{(β1)∗, ϑ2} are con-trollable. Formula (A.3) implies that α =β1+iϑ1ϑ∗1. Using the latter iden-tity, the fact that {β1, ϑ1} is controllable and the equality Imϑ1 = Imϑ1ϑ∗1, we conclude that {α, ϑ1} is controllable too.
It remains to prove that {α, ϑ2} is controllable. Since S0 = In and the quadruple Σ ={α, S0, ϑ1, ϑ2}is admissible, we haveα−α∗ =i(ϑ1ϑ∗1+ϑ2ϑ∗2), and therefore
β1 =α−iϑ1ϑ∗1 =α∗+iϑ2ϑ∗2 = (α−iϑ2ϑ∗2)∗.
Thus α = (β1)∗ +iϑ2ϑ∗2. We know that {(β1)∗, ϑ2} is controllable and that Imϑ2 = Imϑ2ϑ∗2. These facts imply that {α, ϑ2} is also controllable.
Using the identity (A.6) and Theorem A.3 we can show that a proper rational matrix function ϕ also coincides with the second associate function ϕ2,Σ of some admissible quadruple Σ. More precisely, we have the following corollary.
Corollary A.4. Let ϕe be an m2 × m1 rational matrix function. Then ϕe coincides with some function ϕ2,Σ associated with an admissible quadrupleΣ if and only if ϕ is strictly proper. The corresponding Σ ={α, S0, ϑ1, ϑ2} can
be obtained explicitly by using the following procedure. First, we construct a minimal realization of ϕe,
ϕ(z) =e −iθ∗1(zIn+γ)−1θ2.
Next, we choose X to be the unique positive definite solution of the Riccati equation γX−Xγ∗−iXθ1θ∗1X+iθ2θ2∗ = 0. Finally, we put
S0 =In, ϑ1 =X1/2θ1, ϑ2 =X−1/2θ2, (A.19) α=X−1/2γX1/2+iϑ2ϑ∗2. (A.20) Then ϕe=ϕ2,Σ and the pairs {α, ϑ1} and {α, ϑ2} are controllable.
Proof. Put ϕ(z) = ϕ(−z). Thene ϕ is rational, and ϕe is strictly proper if and only if ϕ is strictly proper. But, by Theorem A.3, the latter happens if and only if there exists an admissible quadruple Σ =e {α, S0,ϑe1,ϑe2} such that ϕ(z) = ϕ1,Σe(z). Next, taking into account (A.5), we see that Σ is thee associate quadruple Σ# for Σ = {α, S0,ϑe2,ϑe1}. Then we know from (A.6) that
ϕ(z) =e ϕ(−z) =ϕ1,Σ#(−z) =ϕ2,Σ(z).
Moreover, Theorem A.3 provides a method to constructΣ, which in terms ofe Σ = Σe # (or, equivalently, Σ = Σe#) yields the procedure to recover Σ from
ϕe and formula (A.4).
Proof of Proposition 2.8. Let S(x) be the n×n matrix defined by (2.2), and put
Q(x) =
S0 −iIn e−2ixA
In
0
, x∈R+,
whereAis given by (2.23). In order to prove Proposition 2.8, that is, to prove (2.24), it suffices to show that Q(x) = ∆(x), where ∆(x) =e−ixαS(x)e−ixα∗, for each x ∈ R+. For that we directly differentiate ∆(x) using the first identity in (2.7) (where we substitute the expression for Λ from (2.2)) and obtain
d
dx∆(x) =−i α∆(x) + ∆(x)α∗
+e−2ixαϑ1ϑ∗1−ϑ2ϑ∗2e−2ixα∗. (A.21)
Now, we show thatQsatisfies the same first order linear differential equation Using (2.3) and the definition of A in (2.23) we see that
S0 −iIn
It follows (using again the definiton of A in (2.23)) that i d Equation (A.22) is immediate from Q. Since ∆ and Q satisfy the same first order linear differential equation with the same initial condition ∆(0) = Q(0) =S0, we see that ∆(x) =Q(x) for each x∈R+, as desired.
Acknowledgments. The research of A.L. Sakhnovich was supported by the Austrian Science Fund (FWF) under Grant No. P24301. The authors are grateful to I. Roitberg for her help and useful remarks.
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