1. Show that two matrices in adjacent iterations of the QR eigenvalue algorithm with a single explicit shift, Ak and Ak+1, are orthogonally similar.
2. Suppose A is a symmetric tridiagonal n ×n square matrix.
(a) Describe the nonzero structure of the factors of the QR factorization of A.
(b) Explain how Givens rotations can be used in the computation of the QR factorization of A, and show briefly that the operation count is far below what would be required for a full matrix.
(c) What is the nonzero structure of RQ, and how is this useful for applying the QR iteration for computing eigenvalues?