We propose a Node Centroid method with Hill-Climbing to solve the well-known matrix bandwidth minimization problem, which is to permute rows and columns of the matrix to minimize its bandwidth. Many heuristics have been developed for this NP-complete problem including the Cuthill-McKee (CM) and the Gibbs, Poole and Stockmeyer (GPS) algorithms. Recently, heuristics such as Simulated Annealing, Tabu Search and GRASP have been used, where Tabu Search and the GRASP with Path Relinking have achieved significantly better solution quality than the CM and GPS algorithms. Experimentation shows that the Node Centroid method achieves the best solution quality when compared with these while being much faster than the newly-developed algorithms. Also, the new algorithm achieves better solutions than the GPS algorithm in comparable time.