Right and Left Joint System Representation of a Rational Matrix Function in General Position

Abstract

Listen

Translate article

For a rational k × k matrix function R of one variable in general position, the matrix functions R(x)• R -1 (y) and R -1 (x)• R(y) of two variables are considered. For these matrix functions of two variables, the representations which are analogous to the system representation (or realization) of a rational matrix function of one variable are constructed. This representation of the function R(x)• R -1 (y) (of the function R -1(x) • R(y)) is said to be the joint right (respectively the joint left) system representation of the matrix functions R,R -1. In these representations there appear diagonal n × n matrices, A p = diag(λ1,…,λn) (called the pole matrix for R) and A N = diag (μ 1,…,μ n) (called the zero matrix for R), where λ1,…,λn are poles of R, μ 1,…,μ n are poles of R -1; and k x n matrices F p and F N (called the left pole and zero semi-residual matrices) and n × k matrices G p ,and G N (called the right pole and zero semi-residual matrices) which can be introduced from the additive decompositions \(R(z) = R(\infty ) + {{F}_{\mathcal{P}}}{{(zI - {{A}_{\mathcal{P}}})}^{{ - 1}}}{{G}_{\mathcal{P}}}, {{R}^{{ - 1}}}(z) = R{{(\infty )}^{{ - 1}}} + {{F}_{\mathcal{N}}}{{(zI - {{A}_{\mathcal{N}}})}^{{ - 1}}}{{G}_{\mathcal{N}}}\) The right joint system representation has the form $$R(x) \cdot {{R}^{{ - 1}}}(y) = I + (x - y){{F}_{\mathcal{P}}}{{(xI - {{A}_{\mathcal{P}}})}^{{ - 1}}}{{({{S}^{r}})}^{{ - 1}}}{{(yI - {{A}_{\mathcal{N}}})}^{{ - 1}}}{{G}_{\mathcal{N}}},$$ the left one has the form $${{R}^{{ - 1}}}(x) \cdot R(y) = I + (x - y){{F}_{\mathcal{N}}}{{(xI - {{A}_{\mathcal{N}}})}^{{ - 1}}}{{({{S}^{l}})}^{{ - 1}}}{{(yI - {{A}_{\mathcal{P}}})}^{{ - 1}}}{{G}_{\mathcal{P}}}.$$ The n × n matrices S r and S l(the so-called right and left zero-pole coupling matrices for R) are solutions of the appropriate Sylvester-Lyapunov equations. These matrices are mutually inverse: S r • S l= S l • S r = I. These results are essentially not new: they could be easily derived from known results on realization of a rational matrix functions (for example, from results by L. Sakhnovich or J. Ball, I. Gohberg, L. Rodman), however the method is new, as well as the emphasis on “the left, the right and their relationships”. The presentation is oriented to a “traditional” analyst. No previous knowledge in realization theory of matrix functions or its ideology is assumed. One of the purposes of this paper is to provide a realization theory background for investigations of the deformation theory of a Fuchsian differential system and of rational solutions of the Schlesinger system. As an application we also consider the spectral (Wiener-Hopf) factorization.The concluding Section 5 contains some historical remarks highlighting the role of M.S. Livšic as the forefather of the system realization theory.KeywordsGeneral PositionMatrix FunctionPartial IndexChain IdentitySpectral FactorizationThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.