# Symmetric matrix pencils: codimension counts and the solution of a pair of matrix equations

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## Abstract

The set of all solutions to the homogeneous system of matrix equations (X^{T}A +

AX,X^{T}B + 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. K°agstr¨om, 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.

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