Answer Happy

2. As the first sub-step of the QR algorithm for A, we used an
orthogonal matrix Q1 to reduce the first column a1 to a multiple of
e1. We have seen in class that two following approaches are used.
(i) In the Householder QR algorithm, we choose Q1 to be the
Householder matrix such that Q1a1 = ‖a1‖2e1. (Here without loss of
generality, we assumed a1 is reduced to ‖a1‖2e1 rather than
−‖a1‖2e1.) (ii) In the Givens QR algorithm, we use the product of a
series of Givens rotations Q1 = G(1) n−1 ···G(1) 1 such that Q1a1 =
‖a1‖2e1. Prove that the two Q1’s in (i) and (ii) are different if
all entries of a1 are non-zero. (Actually, there are many other
different ways to orthogonally reduce a1 to a multiple of e1.)
2. As the first sub-step of the QR algorithm for A. we used an orthogonal matrix Q. to reduce the first column ay to a multiple of ei. We have seen in class that two following approaches are used. (i) In the Householder QR algorithm, we choose Q1 to be the Householder matrix such that Qia1 = ||a1|2e1. (Here without loss of generality, we assumed az is reduced to ||a1||2e rather than -||a1||2e1.) (ii) In the Givens QR algorithm, we use the product of a series of Givens rotations Q1 =GA_G such that Qian = |a1||2e1. Prove that the two Qi's in (i) and (ii) are different if all entries of a are non-zero. (Actually, there are many other different ways to orthogonally reduce a to a multiple of e 1.)