If we have a linear system $Ax = b$, and want to solve for $x$ (i.e GEPP), is there a way to solve the system by first computing the Hessenberg reduction of $A$?
2026-03-26 22:19:10.1774563550
Using Hessenburg reduction to solve linear system
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Yes. Reduce $A$ to Hessenberg form $H$, i.e., $H = Q^T A Q$. We have $Ax = b$ if and only if $H y = f$ where $y = Q^T x$ and $f = Q^T b$. Now reduce $H$ to triangular form using Given's rotations, i.e., $T = U^T H$. We have $Hy = f$ if and only if $Ty = g$, where $g = U^T f$. The triangular system can now be solved using substitution.
There are applications where Hessenberg systems occur naturally. A prominent example is the iterative GMRES algorithm for solving linear systems. Here a Hessenberg system is gradually extended and the current system must be solved once during each iteration.
The (two-sided) reduction to Hessenberg form is also the standard pre-processing step for the QR algorithm for computing the eigenvalues of $A$.