# Optimality properties of Galerkin and Petrov-Galerkin methods for linear matrix equations

## TL;DR

This paper investigates the optimality properties of Galerkin and Petrov-Galerkin methods when applied to linear matrix equations, extending their known benefits from numerical analysis to this broader context.

## Contribution

It demonstrates that Galerkin and Petrov-Galerkin methods retain their optimality features for linear matrix equations and introduces constrained minimization techniques within this framework.

## Key findings

- Galerkin and Petrov-Galerkin methods preserve optimality in linear matrix equations
- Novel considerations for applying these methods to general matrix equations
- Proposal of constrained minimization techniques in Petrov-Galerkin methods

## Abstract

Galerkin and Petrov-Galerkin methods are some of the most successful solution procedures in numerical analysis. Their popularity is mainly due to the optimality properties of their approximate solution. We show that these features carry over to the (Petrov-)Galerkin methods applied for the solution of linear matrix equations. Some novel considerations about the use of Galerkin and Petrov-Galerkin schemes in the numerical treatment of general linear matrix equations are expounded and the use of constrained minimization techniques in the Petrov-Galerkin framework is proposed.