CODEN(USA): JSERBR Different Approaches on the Matrix Division and Generalization of Cramer's Rule (original) (raw)

Analogs of Cramer's rule for the least squares solutions of some matrix equations

The least squares solutions with the minimum norm of the matrix equations rmbfArmbfX=rmbfB{\rm {\bf A}}{\rm {\bf X}} = {\rm {\bf B}}rmbfArmbfX=rmbfB, rmbfXrmbfA=rmbfB{\rm {\bf X}}{\rm {\bf A}} = {\rm {\bf B}}rmbfXrmbfA=rmbfB and rmbfArmbfXrmbfB=rmbfD{\rm {\bf A}}{\rm {\bf X}}{\rm {\bf B}} ={\rm {\bf D}} rmbfArmbfXrmbfB=rmbfD are considered in this paper. We use the determinantal representations of the Moore - Penrose inverse obtained earlier by the author and get analogs of the Cramer rule for the least squares solutions of these matrix equations.

Explicit expressions of the generalized inverses and condensed Cramer rules

Linear Algebra and its Applications

In this paper, we obtain an explicit representation of the {2}-inverse A (2) T ,S of a matrix A ∈ C m×n with the prescribed range T and null space S. As special cases, new expressions for the Moore-Penrose inverse A + and Drazin inverse A D are derived. Through explicit expressions, we re-derive the condensed Cramer rules of Werner for minimal-norm least squares solution of linear equations Ax = b and propose two new condensed Cramer rules for the unique solution of a class of singular system Ax = b, x ∈ R(A k ), b ∈ R(A k ), k = Ind(A). Finally, condensed determinantal expressions for A + , A D , AA + , A + A, and AA D are also presented.

Cramer's Rule for Generalized Inverse Solutions of Some Matrices Equations

2015

By a generalized inverse of a given matrix, we mean a matrix that exists for a larger class of matrices than the nonsingular matrices, that has some of the properties of the usual inverse, and that agrees with inverse when given matrix happens to be nonsingular. In theory, there are many different generalized inverses that exist. We shall consider the Moore Penrose, weighted Moore-Penrose, Drazin and weighted Drazin inverses. New determinantal representations of these generalized inverse based on their limit representations are introduced in this paper. Application of this new method allows us to obtain analogues classical adjoint matrix. Using the obtained analogues of the adjoint matrix, we get Cramer's rules for the least squares solution with the minimum norm and for the Drazin inverse solution of singular linear systems. Cramer's rules for the minimum norm least squares solutions and the Drazin inverse solutions of the matrix equations AX = D, XB = D and AXB = D are also obtained, where A, B can be singular matrices of appropriate size. Finally, we derive determinantal representations of solutions of the differential matrix equations, X ′ + AX = B and X ′ + XA = B, where the matrix A is singular.

Analogs of Cramer’s rule for the minimum norm least squares solutions of some matrix equations

Applied Mathematics and Computation, 2012

The least squares solutions with the minimum norm of the matrix equations AX ¼ B; XA ¼ B and AXB ¼ D are considered in this paper. We use the determinantal representations of the Moore-Penrose inverse obtained earlier by the author and get analogs of the Cramer rule for the minimum norm least squares solutions of these matrix equations.

Analogs of Cramer's rule for the least squares solutions

2011

The least squares solutions with the minimum norm of the matrix equations AX = B, XA = B and AXB = D are considered in this paper. We use the determinantal representations of the Moore-Penrose inverse obtained earlier by the author and get analogs of the Cramer rule for the least squares solutions of these matrix equations.

Generalizing Cramer's Rule: Solving Uniformly Linear Systems of Equations

Siam Journal on Matrix Analysis and Applications, 2005

Following Mulmuley's Lemma, this paper presents a generalization of the Moore-Penrose Inverse for a matrix over an arbitrary field. This generalization yields a way to uniformly solve linear systems of equations which depend on some parameters.

Division of two Polynomial Matrices Using the Fundamental Matrix Approach

A new method is presented for carrying out the division of two polynomial matrices which are not necessarily column reduced but have arbitrary form. The given algorithm is iterative, easy to handle and due to its closed matrix form (only constant matrices take part in the algorithm) can be easily implemented in a digital computer using a tool such as Mathematica or Maple.The proposed method can be used in various analysis and synthesis problems such as the evaluation of the impulsive behaviour and the computation of the Controllability/Obsrvability matrices of a linear multivariable system.

Application of the Cramer rule in the solution of sparse systems of linear algebraic equations

Journal of computational and applied mathematics, 2001

In this work, the solution of a sparse system of linear algebraic equations is obtained by using the Cramer rule. The determinants are computed with the help of the numerical structure approach defined in Suchkov (Graphs of Gearing Machines, Leningrad, Quebec, 1983) in which only the non-zero elements are used. Cramer rule produces the solution directly without creating fill-in problem encountered in other direct methods. Moreover, the solution can be expressed exactly if all the entries, including the right-hand side, are ...