The Discussion Of Several Matrix Equation Problems

The constrained matrix equation problem is, in a constrained matrix set, finding a solution of the matrix equation. Different constrained condition and different matrix type will lead to different constrained matrix equation problem. The matrix extension problem is, under some constrained conditions, constructing a matrix A with a given matrix A₀ as its submatrix. On 1979, H.Hochstad first discussed the inverse eigenvalue problem of Jacobi matrix. After that, the matrix extension problem is in great demand.The constrained matrix equation problem and the matrix extension problem have been widely used in structural dynamics, biology, electricity, molecular spectroscopy, control theory, vibration theory, nonlinear program, dynamic analysis and so on, so the discussion of the problems above has actual and applied background.In this paper, we consider several kinds of constrained matrix equation problem and matrix extension problem. The main achievements are as follow:1. The problem of constructing a real symmetric five-diagonal matrix J_n from its defective eigen-pair (λ, X₂), (μ,Y₂), (v,Z₂) and a principle submatrix J_k is discussed. Some necessary and sufficient conditions of solvability are derived. A special real symmetric five-diagonal matrix is considered. A numerical algorithm and a numerical experiment are given.2. The least norm, positive semidefinite and positive definite real solutions of bisymmetric matrix equations (A^TXA, XA — YAD) = (D, 0)are considered by applying the generalized singular value decomposition. The least norm solutions of symmetric and skew antisymmetric matrix equations (A^TXA, XA — YAD) = (D, 0)are also considered.3. The least-squares solutions and the least norm solutions of the quaternion matrix equations AX = B and XD = E are studied by using...