Some Properties And Applications Of Shift Operator

First we utilize the shift operatorS qin linear spaceX qand its properties toget the generalized Jordan’s normal form of a matrix in an arbitrary field F.Andwe could get the conclusion that the generalized Jordan’s normal form of a matrixis unique.Later we use “kernel” and two kinds of defined bilinear forms to get therelationship between the Bezout matrix and the representation ofS qfrom controlbasis to standard basis as well as the relationship between the Hankel matrix andthe representation ofS qfrom standard basis to control basis. From the two specialrelationships we can deduce some beautiful properties of Bezout matrix and Hankelmatrix as well as the relationships between the two kinds of matrices.