Generalized Fiedler pencils with repetition for rational matrix functions

We introduce generalized Fiedler pencil with repetition(GFPR) for an n x n rational matrix function G(?) relative to a realization of G(?). We show that a GFPR is a linearization of G(?) when the realization of G(?) is minimal and describe recovery of eigenvectors of G(?) from those of the GFPRs. In fact, we show that a GFPR allows operation-free recovery of eigenvectors of G(?). We describe construction of a symmetric GFPR when G(?) is symmetric. We also construct GFPR for the Rosenbrock system matrix S(?) associated with an linear time-invariant (LTI) state-space system and show that the GFPR are Rosenbrock linearizations of S(?). We also describe recovery of eigenvectors of S(?) from those of the GFPR for S(?). Finally, We analyze operation-free Symmetric/self-adjoint structure Fiedler pencils of system matrix S(?) and rational matrix G(?). We show that structure pencils are linearizations of G(?).

Symmetric spectral factorisation of self-adjoint rational matrix functions

For a self-adjoint rational matrix function, not necessarily analytic at infinity, the existence of a right (symmetric) spectral factorisation is described in terms of a given left spectral factorisation. The formula for the right spectral factor is given in terms of the formula for the given left spectral factor. All formulas are based on a special realisation of a rational matrix function, which is different from ones that have been used before.

Approximation of Irrational Matrix Functions and Its Application to Continuous-Discrete Model Conversion

In this paper, an improved method of rational approximation is presented for evaluating the irrational matrix function f(A), where A is a square matrix and f(s) is a scalar irrational function which is analytic on the spectrum of A. The improvement in the accuracy of the approximation off (A) by a rational matrix function is achieved by using the multipoint Pade approximants to f(s). An application example to model conversion involving the evaluations of the matrix exponential exp (AT) and the matrix logarithm ln(F) is provided to illustrate the superiority of the method.