The QR Method: A Transformation-Based Algorithm for Eigenvalue Computation

The QR method stands as one of the most efficient techniques for calculating all eigenvalues of general matrices, particularly for small to medium-sized problems. The implementation strategy of the QR algorithm involves decomposing the matrix into an orthogonal matrix Q and an upper triangular matrix R. This decomposition can be achieved through fundamental matrix transformations, which can be interpreted as rotations and scalings performed during the projection of the matrix onto subspace dimensions. From a computational perspective, the QR decomposition is typically implemented using Householder reflections or Givens rotations, with Householder transformations being particularly efficient for dense matrices. The algorithm iteratively applies QR decompositions to converge toward a Schur form where eigenvalues become readily available on the diagonal. The efficiency of the QR method depends on matrix dimensions and condition number, but in most practical scenarios, it demonstrates superior speed and accuracy compared to alternative eigenvalue algorithms. The method finds extensive applications across multiple disciplines including engineering, physics, and computer science, particularly in numerical linear algebra computations and stability analysis problems. Modern implementations often incorporate balancing techniques and implicit shifts to enhance convergence properties for challenging eigenvalue problems.