Browsing by Author "Beavers, Alex Newsom, Jr."
Now showing 1 - 2 of 2
- Results Per Page
- Sort Options
Item A new similarity transformation method for eigenvalues and eigenvectors(1973) Beavers, Alex Newsom, Jr.; Denman, Eugene D.; Ander, Willard N., Jr.; Bargainer, James D., Jr.; Sinkhorn, Richard D.; Tavora, Carlos J.A new method of finding the eigenvalues and eigenvectors of an arbitrary complex matrix is presented. The new method is a similarity transformation method which transforms an arbitrary N x N matrix to a Jordan canonical form in N-l or less transformations. Each transformation matrix is a matrix function— the matrix sign function with a [plus-minus]1 added to the main diagonal elements. Using this matrix function as a similarity transformation gives a block diagonal form which is a reduced form of the transformed matrix. As the Jordan canonical form is found, the eigenvectors are simultaneously found since the product of transformation matrices must be a matrix of eigenvectors. The theoretical development of the new method and a computational scheme with examples are given. In the examples, the computational scheme is applied successfully to matrices, which have characteristics that cause problems for most numerical techniques.Item A solution method for roots of matrix quadratic forms(1972) Beavers, Alex Newsom, Jr.; Denman, Eugene D.; Sinkhorn, Richard D.; Tavora, Carlos J.The algebraic Riccati equation appears in the areas of optimal control, optimal filtering and estimation, and Lyapunov stability studies in a certain form and in the most general form is the generalized matrix quadratic equation. The solution of the algebraic Riccati equation has been obtained in several ways including numerical integration, difference equations, and by computation of the eigenvectors of a matrix composed of the coefficients of the Riccati equation. A new method of computing the solutions to the algebraic Riccati equation is given which is based on the concept of matrix functions. The sign of a matrix is computed to isolate the signs of the real parts of the eigenvalues. Manipulations on this matrix produce what will be called pseudo-Walsh function matrices, i.e., matrices which obey the recursive relations of Walsh functions. The net effect of this procedure is the changing of the signs of particular eigenvalues. The solutions to the algebraic Riccati equation can then be found by the product of a partitioned block and the inverse of a block of a matrix function which is constructed from the pseudo-Walsh matrices. Examples of the method as applied to the general problem and to the particular engineering problems mentioned above will be given.