Rational and Polynomial Matrices

where λ = s or λ = z for a continuousor discrete-time realization, respectively. It is widely accepted that most numerical operations on rational or polynomial matrices are best done by manipulating the matrices of the corresponding descriptor system representations. Many operations on standard matrices (such as finding the rank, determinant, inverse or generalized inverses, nullspace) or the solution of linear matrix equations have natural generalizations for rational matrices. The conjugate transposition of a complex matrix generalizes to the conjugation of a rational matrix, while the full-rank, inner–outer, and spectral factorisations can be seen as generalizations of the familiar LU, QR, and Cholesky factorizations, respectively. Many problems for scalar polynomials and rational functions (poles and zeros, minimum degree or normalized coprime factorizations, and spectral factorization) have nontrivial extensions to polynomial and rational matrices.