The Result of Two Steps of the $LR$ Algorithm is Diagonally Similar to the Result of One Step of the $HR$ Algorithm
Jason Slemons · SIAM Journal on Matrix Analysis and Applications · 2009
Real nonsymmetric tridiagonal matrices arise in various applications. When one is asked to find the eigenvalues of such a matrix, the $QR$ algorithm is used, but this destroys tridiagonal form by converting the matrix to Hessenberg form, resulting in increased storage requirements and numerical operations. The $HR$ algorithm, based on the $HR$ factorization of the matrix into a $(\Delta,\Delta_1)$-orthogonal part H, where $H^T\Delta H=\Delta_1$, and an upper triangular part R, solves this problem. In a result proved by Hongguo Xu, two steps of the $LR$ algorithm are equivalent to one step of the $QR$ algorithm for symmetric matrices. The first object of this paper is to use the $HR$ algorithm to extend Hongguo Xu's result to the nonsymmetric case. Since an $HR$ factorization does not always exist, so we also consider an extension to it called $XHR$ factorization. We then prove a similar result about it.