Several Types Of Matrix Scaling Problem And Several Types Of Matrix Equation

The matrix extension problem is, under some constrained conditions, constructing a matrix A with a given matrix A as its submatrix. The constrained matrix equation problem is, in a constrained matrix set. finding a solution of the matrix equation.The matrix extension problem and constrained matrix equation problem have been widely used in structural design, structural dynamics, biology, electricity, molecular spectroscopy, control theory, vibration theory, nonlinear program, dynamic analysis and so on. This Ph.D. Thesis consider several kinds of matrix extension problems and constrained matrix equation problems. The main achievements are as follow:1. This Ph.D. Thesis firstly considers the Jacobi matrix extension problems constrained by defective eigenpairs and a submatrix, and firstly considers the construction of a Jacobi matrix from its mixed-type eigenpairs. By analysis the structural characterizations of the eigenpairs of the Jacobi matrix, the necessary and sufficient conditions for the existence of and the expressions for the above two problems are derived, and the numerical algorithms and examples to solve the problems are also given.2. This Ph.D. Thesis-firstly considers the real asymmetric, real symmetric, bisym-metric, and symmetric and skew antisymmetric matrix extension problems constrained by the matrix inverse problem AX = B. And also considers, in the solution set, of the corresponding matrix extension problems, the optimal approximation solution to a given matrix A*. The necessary and sufficient conditions for the existence of and the expressions for the above problems are derived, and the numerical algorithm and examples to solve the problems are also given.3. If define inner product on Rn as (x,y) = yTx, then A Rn*n is symmetric (or antisymmetric) matrix if and only if (Ax.y) = (x,Ay) (or (Ax-,y) = - ) for all x,y Rn. According to the relation of above defined inner product, to symmetric (or antisymmetric) matrix, this paper firstly defines the I-general symmetric (or antisymmetric) matrix of satisfying (Ax,y) = (x.Ay) (or (Ax,y) = -) for all x Rn,y R(M) and a given M Rn*p, and the II-general symmetric (or antisymmetric) matrix of satisfying (Ax,y) = (x,Ay) (or (Ax,y) = -{x,Ay}) for all x,y R(M) and a given M Pn*p. As a generalization of symmetric (or antisymmetric) matrix, this paper also defines III-general symmetric (or antisymmetric) matrix of satisfying MAN = (MAN)T (or MAN = -(MAN)T) for given M Pn*p and N Rq*n. By analysis the structural characterizations of the above three kindsof matrices together with using the singular-value, generalized singular-value decompositions of the matrix pair, the necessary and sufficient conditions for the existence of and the expressions for the nearest solution of matrix inverse problems AX = B for the above three kinds of matrices to a given matrix A* are derived, and the numerical algorithm and examples to solve the problems are also given.4. Let P Rn*n and Q Rm*m are selfadjoint involutory matrices, i.e., PH = P, QH = Q, P2 = In, Q2 = Im. An n * m complex matrix A is said to be a generalized reflexive (or anti-reflexive) matrix with respect to the selfadjoint involutory matrices P and Q if A = PAQ (or A = -PAQ). A generalized reflexive (or anti-reflexive) matrix with respect to the selfadjoint involutory matrices P and Q is a generalization of centrosymmetric (or centroantisymmetric) matrix. This paper establishes the necessary and sufficient conditions for the existence of and the expressions for the nearest generalized reflexive (or anti-reflexive) solution of matrix inverse problem AX ~ B or least-squares problem ||AX -B|| = min to a given matrix A*. In addition, the numerical algorithm and examples to solve the pioblems are also given.5. Using the generalized singular-value decompositions of the matrix pair, this paper considers the nearest bisymmtcric, biantisymnietric, symmetric and skew antisymmetric, and antisymmetric and skew symmetric solutions of matrix eq...