Ela Symmetric Matrix Pencils: Codimension Counts and the Solution of a Pair of Matrix Equations∗
نویسندگان
چکیده
The set of all solutions to the homogeneous system of matrix equations (XA + AX,XB +BX) = (0,0), where (A,B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A,B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. K̊agström, and V.V. Sergeichuk. Skew-symmetric matrix pencils: Codimension counts and the solution of a pair of matrix equations. Linear Algebra Appl., 438:3375–3396, 2013.], where the corresponding problems for skew-symmetric matrix pencils are solved. The new results will be useful in the development of the stratification theory for orbits of symmetric matrix pencils.
منابع مشابه
Symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations
The set of all solutions to the homogeneous system of matrix equations (XA + AX,XB +BX) = (0,0), where (A,B) is a pair of symmetric matrices of the same size, is characterized. In addition, the codimension of the orbit of (A,B) under congruence is calculated. This paper is a natural continuation of the article [A. Dmytryshyn, B. K̊agström, and V.V. Sergeichuk. Skew-symmetric matrix pencils: Codi...
متن کاملThe (R,S)-symmetric and (R,S)-skew symmetric solutions of the pair of matrix equations A1XB1 = C1 and A2XB2 = C2
Let $Rin textbf{C}^{mtimes m}$ and $Sin textbf{C}^{ntimes n}$ be nontrivial involution matrices; i.e., $R=R^{-1}neq pm~I$ and $S=S^{-1}neq pm~I$. An $mtimes n$ complex matrix $A$ is said to be an $(R, S)$-symmetric ($(R, S)$-skew symmetric) matrix if $RAS =A$ ($ RAS =-A$). The $(R, S)$-symmetric and $(R, S)$-skew symmetric matrices have a number of special properties and widely used in eng...
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 کاملEla 450
We solve the matrix equation XA+AX = 0, where A ∈ C is an arbitrary given square matrix, and we compute the dimension of its solution space. This dimension coincides with the codimension of the tangent space of the congruence orbit of A. Hence, we also obtain the (real) dimension of congruence orbits in C. As an application, we determine the generic canonical structure for congruence in C and a...
متن کاملPreconditioned Generalized Minimal Residual Method for Solving Fractional Advection-Diffusion Equation
Introduction Fractional differential equations (FDEs) have attracted much attention and have been widely used in the fields of finance, physics, image processing, and biology, etc. It is not always possible to find an analytical solution for such equations. The approximate solution or numerical scheme may be a good approach, particularly, the schemes in numerical linear algebra for solving ...
متن کامل