MATRIX SHAPES INVARIANT UNDER THE SYMMETRICAL QR ALGORITHM

The QR algorithm is a basic algorithm for computing the eigenvalues of dense matrices. For efficiency reasons it is prerequisite that the al gorithm is applied only after the original matrix has been reduced to a matrix of a particular shape, most notably Hessenberg and tridiagona l, which is preserved during the iterative process. In certain circums tances a reduction to another matrix shape may be advantageous. In thi s paper, we identify which zero patterns of symmetric matrices are pre served under the QR algorithm.