The Rayleigh Quotient Iteration for Non-Normal Matrices.

Abstract : The Rayleigh Quotient Iteration (RQI) is a method for computing eigenvectors and eigenvalues of a square matrix. The behaviour, both local and global, of RQI with symmetric and normal matrices is almost completely understood. The vector sequence converges for almost all starting vectors. In this paper, the author investigates the global properties of RQI on non-normal matrices. Results on nearly normal matrices with real eigenvalues are obtained, and at the other extreme, results on completely degenerate matrices are also obtained. In particular, the question of global convergence of the vector iteration on a general matrix is reduced to the convergence of the scalar sequence of the Rayleigh quotients. In practice, the vector iteration always converges. The main difficulty in the degenerate case is that the iteration function is discontinuous near the eigenvector. An example is used to display the sectorial behaviour of the iteration. Further the author constructs a sequence of numbers that converges to the eigenvalue. Yet if the numbers are used as shifts with inverse iteration, the vector sequence fails to converge.