Please help on these Let u be a nonzero vector in Rn. Let H = I - 2p uuT where p = 1/(uTu). H is called a Householder matrix. Prove that H is orthogonal. Let x be an arbitrary vector in Rn and let y = Ax. Prove that ||x|| = \\\\y\\\\ if and only if A is orthogonal. Here ||.|| is the usual Euclidean norm. Let Q1 , Q2 be two n x n orthogonal matrices. Prove or disprove that Q = Q1Q2 is orthogonal. Solution 7) H^T = I -2p (uu^T)^T = I -2puu^T = H , and H.H^T = (I-2p uu^T)(I-2p uu^T) = I-4puu^T+4p^2 u(u^Tu)u^T = I - 4p uu^T + 4p^2u*(1/p)*u^T = I So H is orthogonal 8) ||x||=||y|| <=> ||Ax||=||x|| <=> <Ax,Ax> = <x,x> <=> A is orthogonal 9) QQ^T = Q1Q2(Q1Q2)^T = Q1(Q2Q2^T)Q1^T = Q1*I*Q1^T=Q1Q1^T=I So Q is orthogonal .

1. Please help on these
Let u be a nonzero vector in Rn. Let H = I - 2p uuT where p = 1/(uTu). H is called a Householder
matrix. Prove that H is orthogonal. Let x be an arbitrary vector in Rn and let y = Ax. Prove that
||x|| = y if and only if A is orthogonal. Here ||.|| is the usual Euclidean norm. Let Q1 , Q2 be
two n x n orthogonal matrices. Prove or disprove that Q = Q1Q2 is orthogonal.
Solution
7) H^T = I -2p (uu^T)^T = I -2puu^T = H , and
H.H^T = (I-2p uu^T)(I-2p uu^T) = I-4puu^T+4p^2 u(u^Tu)u^T = I - 4p uu^T +
4p^2u*(1/p)*u^T = I
So H is orthogonal
8)
||x||=||y|| <=> ||Ax||=||x|| <=> <Ax,Ax> = <x,x> <=> A is orthogonal
9)
QQ^T = Q1Q2(Q1Q2)^T = Q1(Q2Q2^T)Q1^T = Q1*I*Q1^T=Q1Q1^T=I
So Q is orthogonal