A matrix in the Schwarz block form and the stability of matrix polynomials

The Schwarz matrix was established by H. R. Schwarz in 1956. He used several elementary transformations to transform a given system matrix to the Schwarz matrix. Since then numerous authors have investigated the properties and applications of the Schwarz matrix. Also, various transformation matrices which relate a given system matrix to the Schwarz matrix have been established. However, most of the earlier developed transformation matrices were too complicated to implement, and also they were restricted to single variable systems only. In this research a matrix which consists of block elements is established in the Schwarz block form via a linear transformation. A new block-transformation matrix is established for transforming the companion block form to the Schwarz block form. A sufficient condition has been derived for determining the stability of a multivariable system whose characteristics are expressed by a polynomial matrix. At the same time, to determine the stability of multivariable systems, the direct extension of the well-known scalar Routh theorem to the matrix Routh theorem has also been studied in this research.