Minimal Factorization of Matrix and Operator Functions

Bart, Harm; Gohberg, Israel; Kaashoek, M. A.

doi:10.1007/978-3-0348-6293-6

Cited by 448 publications

(300 citation statements)

References 0 publications

Supporting

Mentioning

285

Contrasting

Unclassified

Order By: Relevance

“…We note that ]Ct, ]Cr, and s do not have eigenvalues on the imaginary axis. This follows from the invertibility of I~ -R(A) R(),) t and Corollary 2.7 in [13]; for/Q and Er this also follows immediately from the special form of the matrices ]Q and ]C~ in (7.4) and the fact that A has no eigenvalues on the imaginary axis. Hence the matrices (~ -i]Cl) -1, (A -iK:~) -1, and (A -ig) -1 in (7.2), (7.3), (7.5), and (7.6) all exist for A e R.…”

Section: Tr(a)-i [Tr(a*)t]-l=in-ieo Bt](a-ie)-i [~]mentioning

confidence: 83%

“…[8,27]). When the reflection coefficients are rational, we apply state space methods [13] to solve the Marchenko equations and the inverse problem explicitly. For rational reflection coefficients, this approach provides us with a systematic inversion method for inverse scattering problems on the line, which is different from previous methods such as those used in [9].…”

Section: Dz(xa) [ P(x) Ai~ -V(x) L Z(xa) ' Dx -;Z~ -V(x) -P(x) P(mentioning

confidence: 99%

“…From the theory of transfer functions [13], since R(A) -+ 0 as t --+ ~oo, it follows that R(A) can be represented in the form…”

Section: Construction Of the Scattering Matrixmentioning

confidence: 99%

See 2 more Smart Citations

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Aktosun

Klaus

Mee

2000

Integr equ oper theory

View full text Add to dashboard Cite

A direct and inverse scattering theory on the full line is developed for a class of firstorder selfadjoint 2n • 2n systems of differential equations with integrable potential matrices. Various properties of the corresponding scattering matrices including unitarity and canonical Wiener-Hopf factorization are established. The Marchenko integral equations are derived and their unique solvability is proved. The unique recovery of the potential from the solutions of the Marchenko equations is shown. In the case of rational scattering matrices, state space methods are employed to construct the scattering matrix from a reflection coefficient and to recover the potential explicitly.

show abstract

Section: Tr(a)-i [Tr(a*)t]-l=in-ieo Bt](a-ie)-i [~]mentioning

confidence: 83%

Section: Dz(xa) [ P(x) Ai~ -V(x) L Z(xa) ' Dx -;Z~ -V(x) -P(x) P(mentioning

confidence: 99%

See 1 more Smart Citation

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Aktosun

Klaus

Mee

2000

Integr equ oper theory

View full text Add to dashboard Cite

show abstract

“…In this section we prove a number of results on observable, controllable and minimal GR-nodes in the multivariable non-commutative setting, which generalize some well known statements for one-variable nodes (see [15]). Let us introduce the k-th truncated observability matrix O k and the k-th truncated controllability matrix C k of a GR-node (2.6) by…”

Section: Preliminariessupporting

confidence: 60%

“…Moreover pr Iγ −Π (α) is similar to β ′ , and pr Π (α) is similar to β ′′ . The uniqueness of Π is proved in the same way as in [15,Theorem 4.8]. The uniqueness of the GR-node similarity follows from Theorem 3.5.…”

Section: 1])mentioning

confidence: 84%

Matrix-J-unitary Non-commutative Rational Formal Power Series

Alpay

Kalyuzhnyĭ-Verbovetzkiĭ

The State Space Method Generalizations and Applications

View full text Add to dashboard Cite

Abstract. Formal power series in N non-commuting indeterminates can be considered as a counterpart of functions of one variable holomorphic at 0, and some of their properties are described in terms of coefficients. However, really fruitful analysis begins when one considers for them evaluations on N -tuples of n × n matrices (with n = 1, 2, . . .) or operators on an infinite-dimensional separable Hilbert space. Moreover, such evaluations appear in control, optimization and stabilization problems of modern system engineering.In this paper, a theory of realization and minimal factorization of rational matrix-valued functions which are J-unitary on the imaginary line or on the unit circle is extended to the setting of non-commutative rational formal power series. The property of J-unitarity holds on N -tuples of n × n skew-Hermitian versus unitary matrices (n = 1, 2, . . .), and a rational formal power series is called matrix-J-unitary in this case. The close relationship between minimal realizations and structured Hermitian solutions H of the Lyapunov or Stein equations is established. The results are specialized for the case of matrix-J-inner rational formal power series. In this case H > 0, however the proof of that is more elaborated than in the one-variable case and involves a new technique. For the rational matrix-inner case, i.e., when J = I, the theorem of Ball, Groenewald and Malakorn on unitary realization of a formal power series from the non-commutative Schur-Agler class admits an improvement: the existence of a minimal (thus, finite-dimensional) such unitary realization and its uniqueness up to a unitary similarity is proved. A version of the theory for matrix-selfadjoint rational formal power series is also presented. The concept of non-commutative formal reproducing kernel Pontryagin spaces is introduced, and in this framework the backward shift realization of a matrix-J-unitary rational formal power series in a finite-dimensional non-commutative de Branges-Rovnyak space is described. Mathematics Subject Classification (2000). Primary 47A48; Secondary 13F25, 46C20, 46E22, 93B20, 93D05.

show abstract

Double broadband matching and the problem of reciprocal reactance 2n‐port cascade decomposition

Youla

Carlin

Yarman

1984

Circuit Theory & Apps

View full text Add to dashboard Cite

SUMMARYLet N, N l , N2 and N3 be prescribed reciprocal reactance 2n-ports. Then, under certain mild restrictions, this paper supplies answers to the following two related problems:PI. Find the necessary and sufficient conditions for the physical extractability of Nl from the front-end of N. Pz. Given that N2 and N3 are individually physically extractable from the back and front-ends of N, respectively,The criteria are formulated in terms of the associated scattering matrices and are reasonably simple to apply. Moreover, they also have a clear-cut network significance involving transmission zeros. To illustrate their use, a recent result for the design of non-degenerate double broadband-matching equalized is generalized to a 2n-port setting in Theorem 2, corollary 2.Lastly, to round out the development, impedance versions of both Theorem 1 and Theorem 2, corollary 2, are presented in Section 2. This restatement is accomplished with the aid of a new Darlington 2n-port embedding for passive reciprocal n-ports that is phrased entirely in the language of impedance matrices. find a set of sharp sufficient conditions for their simulations physical extractability from N.

show abstract

Minimal Factorization of Matrix and Operator Functions

Cited by 448 publications

References 0 publications

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Matrix-J-unitary Non-commutative Rational Formal Power Series

Double broadband matching and the problem of reciprocal reactance 2n‐port cascade decomposition

Contact Info

Product

Resources

About