QR algorithm

Summary

The QR algorithm computes all the eigenvalues of a square matrix. It repeatedly factors the matrix as $QR$ by the QR decomposition and multiplies the factors in the reverse order, $RQ$. Each step is a similarity by an orthogonal matrix, so the eigenvalues do not change. The matrices tend to an upper triangular form with the eigenvalues on the diagonal. With shifts and a preliminary reduction to Hessenberg form, it is the standard method for dense eigenvalue problems.