The matrix equation \(XA=A^ TX\) and an associated algorithm for solving the inertia and stability problems (Q1097212)

This paper considers the matrix equations (1) \(XA=A^ TX\) with A as an unreduced lower Hessenberg matrix of order n, and establishes an interesting relationship between a given solution X of (1) and the associated matrix polynomial p(A). It is used to develop an algorithm for computing the inertia of A. The inertia of a matrix is defined to be an integer triple \(In(A)=(\pi (A),\nu (A),\delta (A))\), where \(\pi\) (A), \(\nu\) (A) and \(\delta\) (A) are respectively the numbers of eigenvalues of A with positive, negative, and zero real parts. This paper proposes an algorithm for computing In(A) and compares its efficiency with existing ones.