Some Generalized Inverse Eigenvalue Problems For Jacobi Matrix With Special Structures And Its Application

The generalized inverse eigenproblem concerns the reconstruction of matrices in a constr aint matrix set which is from prescribed complete /partial information of generalizedized eige nvalues or eigenvectors and from partial submatrices or elements. Different classes of matrix sets imply different kinds of generalized inverse eigenproblems. A generalized inverse eigen problem arises in a remarkable variety of application,such as structural design, parameter ide ntification, main component analysis, structure dynamics, molecule spectroscopy, oscillation theory, finite element theory etc. The theory of the generalized inverse eigenproblem is just prompted to develop quickly by many different questions arisen in those fields. And it has bec ome one of the most flourish and popular research problems in numerical algebra. The main works and results of this dissertation are as follows.1.The generalized inverse eigenproblem of the first class of Jacobi matrices with special structure (which is also called fixed-fixed Jacobi matrix) and its applications are studied. Thro ugh discussing the 2x2 partitioned form of fixed-fixed Jacobi matrices,we put forward that the generalized inverse eigenproblems with unconstraint, with orthogonal constraint, with total m ass constraint of system or with energy constraint, as well as the optimal structural design pro- blem of the fixed-fixed mass - spring system with optimizing total mass of the system or ener- gy. The solvable conditions and the general expressions for the solutions are obtained .Further more, the numerical algorithms and some examples to solve the problems above are also obta- ined.2. The generalized inverse eigenproblem of the second class of Jacobi matrices with spe cial structure (which is also called fixed-free Jacobi matrix) and its applications are studied. Through discussing the 3x3 partitioned form of fixed-free Jacobi matrices, we put forward th- at the generalized inverse eigenproblems with unconstraint, with orthogonal constraint, with total mass constraint of system or with energy constraint, as well as the optimal structural des ign problem of the fixed-free mass - spring system with optimizing total mass of the system or energy. The solvable conditions and the general expressions for the solutions are obtained . Moreover, the numerical algorithms and examples to solve the problems above are also obta ined.3. The generalized inverse eigenproblem of the third class of Jacobi matrices with spe cial structure (which is also called periodic Jacobi matrix) and its applications are studied. Through discussing the 2x2 partitioned form of periodic Jacobi matrices, we put forward that the generalized inverse eigenproblem with unconstraint . The solvable conditions and the gen eral expressions for the solution is obtained . Furthermore, the numerical algorithm and exa mple to solve the problems above are also obtained.4. The generalized inverse eigenproblem of the fourth class of Jacobi matrices with sp ecial structure (which is also called integer Jacobi matrix) and its applications are studied. Through discussing the 2x2 and 3x3 partitioned form of integer Jacobi matrices, we put for ward that the generalized inverse eigenproblems in finite fields GF(p) . The solvable coniti ons and the general expressions for the solutions are obtained . Furthermore, it is considered that the application of integer Jacobi matrices in HILL cryptosystem. The numerical algori- thm and example to solve the above problems are also obtained. New HILL cryptosystem has the effect of one-time-pad and the digital signature function.5. Two kinds of the reconstruction problems for the mass-spring system under nature frequencies from its subsystem in order constraint are considered. The solvable conditions and the general expressions for the solution of the problem above are obtained.6. Making use of the singular value decomposition, the generalized singular value dec omposition ,and the generalized inverse , we study the expansion and optimal approximat ions problems of matrices having 3x3 partitioned forms under the nonleading sub-matrix constraint, and obtain the necessary and sufficient conditions for the existence of and the general expressions for the solution of the problem above.