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

Cramer’s Rules for the System of Two-Sided Matrix Equations and of Its Special Cases

Matrix Theory - Applications and Theorems, 2018

Within the framework of the theory of row-column determinants previously introduced by the author, we get determinantal representations (analogs of Cramer's rule) of a partial solution to the system of two-sided quaternion matrix equations A 1 XB 1 =C 1 , A 2 XB 2 =C 2. We also give Cramer's rules for its special cases when the first equation be one-sided. Namely, we consider the two systems with the first equation A 1 X=C 1 and XB 1 =C 1 , respectively, and with an unchanging second equation. Cramer's rules for special cases when two equations are one-sided, namely the system of the equations A 1 X=C 1 , XB 2 =C 2 , and the system of the equations A 1 X=C 1 , A 2 X=C 2 are studied as well. Since the Moore-Penrose inverse is a necessary tool to solve matrix equations, we use its determinantal representations previously obtained by the author in terms of row-column determinants as well.

Old and new proofs of Cramer's rule

Applied Mathematical Sciences, 2014

In spite of its high computational cost, Cramer's Rule for solving systems of linear equations is of historical and theoretical importance. In this paper we list six different proofs of it, the last of which has not apparently been published elsewhere. A discussion on their educational value and the tools involved is also included.

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.

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.