The Study Of Bezout Matrix Pencil Over An Arbitrary Field

Many scholars have long payed attention to the study of Bezout matrixbecause of its important role in system stability theory. In this thesis, we usealgebric method to investigate the Bezout matrix pencil over an arbitrary field.With the help of the properties of the block diagonalization of Bezout matrix andcompanion matrix, and the properties of the minimal polynomial annihilatingJordan block as well as the linearity of Bezout matrices, we obtain some results onthe polynomial Bezout matrix pencil with respect to a general basis andToeplitz-Bezout matrix pencil with respect to the standard power basis over analgebraically closed field. The Bezout matrix pencil with respect to a Jacobsonchain basis over non-algebraically closed field is also investigated.The paper is divided into four chapters, the principal contents are as follows:In chapter one, the backgrounds and present situation of Bezout matrix pencilas well as the main work of our thesis are introduced.In Chapter two, we prove that over an algebraically closed field the linearcombinations of the product of polynomial Bezoutians with arbitrary nonnegativeinteger power of the transpose of confederate companion matrix is also apolynomial Bezout matrix, and we obtain their generating functions. The concept ofpolynomial Bezoutian pencil is introduced and the contact between polynomialBezoutian pencil and polynomial Bezout matrix space is established. To verify theresult some numerical examples are given Finally, the results are extended toToeplitz-Bezout matrix pencil.In Chapter three, applying the known conclusion of Jacobson chain-basedBezout matrix over the arbitrary field, polynomial Bezout matrix pencil isgeneralized to Bezout matrix pencil with respect to a Jacobson chain basis. Usingthe similar method of polynomial Bezout matrix pencil over the non-algebraicallyclosed field, we get some similar results for such matrices.In chapter four, the main work of the paper are summarized and some prolemsto be resolved in the future are put forward.