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 ej. We have seen in class that two following approaches are used. 1) (i) In the Householder QR algorithm, we choose Q1 to be the Householder matrix such that Q1a1 || 21||2e1. (Here without loss of generality, we assumed ai is reduced to ||a1||2e rather than -||a1||201.) (ii) In the Givens QR algorithm, we use the product of a series of Givens rotations Q1 G42_,...G) such that Qia1 = ||a1||2e1. Prove that the two Qs in (i) and (ii) are different if all entries of a are non-zero. (Actually, there are many other different ways to orthogonally reduce aj 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 ej. We have seen in class that two following approaches are used. 1) (i) In the Householder QR algorithm, we choose Q1 to be the Householder matrix such that Q1a1 || 21||2e1. (Here without loss of generality, we assumed ai is reduced to ||a1||2e rather than -||a1||201.) (ii) In the Givens QR algorithm, we use the product of a series of Givens rotations Q1 G42_,...G) such that Qia1 = ||a1||2e1. Prove that the two Qs in (i) and (ii) are different if all entries of a are non-zero. (Actually, there are many other different ways to orthogonally reduce aj to a multiple of e1.)