A description of the QR Algorithm to compute the eigenvalues of a matrix A ∈ â„â¿Ë£â¿ is given below without shifts.
Algorithm 1: The QR Algorithm without Shifts
Aâ‚€ = A
for k = 0, 1, 2, ... do
Compute a QR factorization Aâ‚– = Qâ‚–â‚Šâ‚Râ‚–â‚Šâ‚
Aâ‚–â‚Šâ‚ = Râ‚–â‚Šâ‚Qâ‚–â‚Šâ‚
end for
Here, you are expected to establish the equivalence of the QR algorithm given above and the unnormalized simultaneous power iteration given below. Assume throughout that A is invertible.
Algorithm 2: Unnormalized Simultaneous Power Iteration
for k = 1, 2, ... do
Compute a QR factorization A = QRâ‚–
A = Râ‚–Qâ‚–
end for
For both algorithms, assume also QR factorizations are computed so that the diagonal entries of the upper triangular R factor are all positive. In this case, the QR factorization is unique, i.e., for every invertible C ∈ â„â¿Ë£â¿ there is only one QR factorization of the form C = QR where Q ∈ â„â¿Ë£â¿ is orthogonal, and R ∈ â„â¿Ë£â¿ is upper triangular with positive entries along the diagonal.
Prove the following:
(i) A QR factorization for A is given by
A = Qâ‚Qâ‚‚...Qâ‚–Râ‚Râ‚‚...Râ‚–
for all k such that k ≥ 1.
IS[BOJV=X Hint: First prove Aâ‚– = [Qâ‚–]áµ€...[Qâ‚]áµ€AQâ‚Qâ‚–Vâ‚–, k ≥ 1.)