The Structure Analysis Of Semiseparable Matrices And QR Iterations

The semiseparable matrix is one class of matrix with special matrix struc-ture.Especially,the symmetric semiseparable matrix can be represented by two vectors.The semiseparable matrix appears when we deeply study the in-verse of tridiagonl matrix,The class of semiseparable matrics has important connections with the class of tridiagonal matrices.It is the well known prop-erties,which the inverse of a nonsingular,irreducible symmetric tridiagonal matrix is a symmetric semiseparable matrix.Moreover,there are more similar conclusions on some theories.For example,the Implicit-Q theorem for the semiseparable matrix and tridiagonal matrix is absolutely similar.And their matrix structure is invariant under QR iterations.Firstly,we analyse the structure of semiseparable matrices.We also state the definitions of semiseparable matrices and some relative matrices.Further more,we prove a theorem which can identify the semiseparable matrices.It is well known any symmetric matrix can be reduced by an orthogonal similarity transformation into tridiagonal form.Similarly,we showthat the orthogonal matrix appearing in the QR factorization of a suitable defined inverted Krylov matrix transforms a symmetric matrix into a semiseparable matrix.We not only show the link between the inverted Krylov matrix and the semiseparable matrix but also prove the result.Moreover,we discuss how the semiseparable matrix is irreducible.We also study the necessary and sufficient conditions ensuring this case.Next,we give some theoretical structural properties of the QR factorization of the semiseparable matrix.We analysis the structure of the orthogonal matrix and the upper-triangular matrix.And then we study the Implicit-Q theorem for the semiseparable matrices.In particular,we give a proof by using their inverses which are tridiagonal matrices.Finally,we study the QR iterations of the semiseparable matrix and the diagonal plus semiseparabte matrix.And we prove that their structures are invariant under QR iterations by adequately using previous results.