Extending Four Displacement Principles to Solve Matrix Equations
نویسنده
چکیده
For xed matricesM and N , the linear transformation A 7! A MAN is called a displacement of the matrix A. Displacements can simplify matrix equations, as well as matrices themselves. Four principles to solve matrix equations are identi ed: 1) a variety of displacements are needed, 2) we want to recover a matrix from its displacement, 3) changing M and N is natural when A is transposed or inverted, and 4) formulas for displacements of matrix products are required. All four principles are extended in this paper. These extensions are illustrated using the class of Krylov matrices, which includes circulant, Vandermonde, Toeplitz, Hankel and other structured matrices as simple special cases.
منابع مشابه
First Principles Derivation of Displacement and Stress Function for Three-Dimensional Elastostatic Problems, and Application to the Flexural Analysis of Thick Circular Plates
In this study, stress and displacement functions of the three-dimensional theory of elasticity for homogeneous isotropic bodies are derived from first principles from the differential equations of equilibrium, the generalized stress – strain laws and the geometric relations of strain and displacement. It is found that the stress and displacement functions must be biharmonic functions. The deriv...
متن کاملA numerical algorithm for solving a class of matrix equations
In this paper, we present a numerical algorithm for solving matrix equations $(A otimes B)X = F$ by extending the well-known Gaussian elimination for $Ax = b$. The proposed algorithm has a high computational efficiency. Two numerical examples are provided to show the effectiveness of the proposed algorithm.
متن کاملON MAXWELL'S STRESS FUNCTIONS FOR SOLVING THREE DIMENSIONAL ELASTICITY PROBLEMS IN THE THEORY OF ELASTICITY
The governing equations of three dimensional elasticity problems include the six Beltrami-Michell stress compatibility equations, the three differential equations of equilibrium, and the six material constitutive relations; and these are usually solved subject to the boundary conditions. The system of fifteen differential equations is usually difficult to solve, and simplified methods are usual...
متن کاملOn the numerical solution of generalized Sylvester matrix equations
The global FOM and GMRES algorithms are among the effective methods to solve Sylvester matrix equations. In this paper, we study these algorithms in the case that the coefficient matrices are real symmetric (real symmetric positive definite) and extract two CG-type algorithms for solving generalized Sylvester matrix equations. The proposed methods are iterative projection metho...
متن کاملWave propagation in a transversely isotropic microstretch elastic solid
Background: The theory of microstretch elastic bodies was first developed by Eringen (1971, 1990, 1999, 2004). This theory was developed by extending the theory of micropolar elastcity. Each material point in this theory has three deformable directors. Methods: The governing equations of a transversely isotropic microstretch material are specialized in x-z plane. Plane wave solutions of these g...
متن کامل